Abstract

This paper presents a novel algorithm for the partial reconstruction of interference pattern envelopes. In multi-pulse train interferometers, the exact determination of the peak position of the envelope of interference fringes is of paramount importance. The estimation of the interference pattern envelope usually involves the use of discrete Fourier transform (DFT). The proposed algorithm is based on the chirp Z-transform (CZT) instead of DFT and avoids estimating the entire envelope of the interference pattern. It is sufficient for determining part of the envelope around the peak value position. The proposed approach is presented and illustrated for the first time by means of optical fringes. The experimental results demonstrate that this approach is reliable for partial envelope determination.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009).
    [Crossref] [PubMed]
  2. D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Time-of-flight method using multiple pulse train interference as a time recorder,” Opt. Express 19(6), 4881–4889 (2011).
    [Crossref] [PubMed]
  3. M. Takeda, “Fourier Fringe Demodulation”, in Phase Estimation in Optical Interferometry (CRC Press, 2014), pp. 1–30.
  4. L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Labs Tech. 48(5), 1249–1292 (1969).
    [Crossref]
  5. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29(14), 1668–1670 (2004).
    [Crossref] [PubMed]
  6. T. T. Wang, “The segmented chirp Z-transform and its application in spectrum analysis,” IEEE Trans. Instrum. Meas. 39(2), 318–323 (1990).
    [Crossref]
  7. N. Q. Ngo, “Optical chirp z-transform processor with a simplified architecture,” Opt. Express 22(26), 32329–32343 (2014).
    [Crossref] [PubMed]
  8. X.-G. Xia, “Discrete chirp-Fourier transform and its application to chirp rate estimation,” IEEE Trans. Signal Process. 48(11), 3122–3133 (2000).
    [Crossref]
  9. J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
    [Crossref] [PubMed]
  10. X. Wang, S. Takahashi, K. Takamasu, and H. Matsumoto, “Space position measurement using long-path heterodyne interferometer with optical frequency comb,” Opt. Express 20(3), 2725–2732 (2012).
    [Crossref] [PubMed]
  11. S. A. Shilling, A study of the chirp Z-tranform and its applications, ((Unpublished doctoral dissertation), Kansas State University, Manhattan, KS., 1972).
  12. B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom Algorithms for Digital Holography”, in Information Optics and Photonics: Algorithms, Systems, and Applications, B. Javidi and T. Fournel, eds. (Springer New York, New York, NY, 2010).
  13. D. Wei and M. Aketagawa, “Automatic selection of frequency domain filter for interference fringe analysis in pulse-train interferometer,” Opt. Commun. 425, 113–117 (2018).
    [Crossref]
  14. J. W. Goodman, Introduction to Fourier optics, 2nd ed., McGraw-Hill series in electrical and computer engineering (McGraw-Hill, New York, 1996).
  15. A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics (Cambridge University, 1996).
  16. D. Wei and M. Aketagawa, “Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement,” Opt. Commun. 382, 604–609 (2017).
    [Crossref]
  17. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (2nd ed.) (Prentice-Hall, Inc., 1999).
  18. G. D. Martin, Chirp Z-transform spectral zoom optimization with MATLAB, (University of North Texas, 2019).

2018 (1)

D. Wei and M. Aketagawa, “Automatic selection of frequency domain filter for interference fringe analysis in pulse-train interferometer,” Opt. Commun. 425, 113–117 (2018).
[Crossref]

2017 (1)

D. Wei and M. Aketagawa, “Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement,” Opt. Commun. 382, 604–609 (2017).
[Crossref]

2014 (1)

2012 (1)

2011 (1)

2009 (1)

2006 (1)

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

2004 (1)

2000 (1)

X.-G. Xia, “Discrete chirp-Fourier transform and its application to chirp rate estimation,” IEEE Trans. Signal Process. 48(11), 3122–3133 (2000).
[Crossref]

1990 (1)

T. T. Wang, “The segmented chirp Z-transform and its application in spectrum analysis,” IEEE Trans. Instrum. Meas. 39(2), 318–323 (1990).
[Crossref]

1969 (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Labs Tech. 48(5), 1249–1292 (1969).
[Crossref]

Aketagawa, M.

D. Wei and M. Aketagawa, “Automatic selection of frequency domain filter for interference fringe analysis in pulse-train interferometer,” Opt. Commun. 425, 113–117 (2018).
[Crossref]

D. Wei and M. Aketagawa, “Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement,” Opt. Commun. 382, 604–609 (2017).
[Crossref]

Balcom, B.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Dierkes, T.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Halse, M.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Kaffanke, J.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Leach, M. O.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Matsumoto, H.

Ngo, N. Q.

Rabiner, L. R.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Labs Tech. 48(5), 1249–1292 (1969).
[Crossref]

Rader, C. M.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Labs Tech. 48(5), 1249–1292 (1969).
[Crossref]

Rioux, J.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Romanzetti, S.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Schafer, R. W.

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Labs Tech. 48(5), 1249–1292 (1969).
[Crossref]

Shah, N. J.

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Takahashi, S.

Takamasu, K.

Wang, T. T.

T. T. Wang, “The segmented chirp Z-transform and its application in spectrum analysis,” IEEE Trans. Instrum. Meas. 39(2), 318–323 (1990).
[Crossref]

Wang, X.

Wei, D.

D. Wei and M. Aketagawa, “Automatic selection of frequency domain filter for interference fringe analysis in pulse-train interferometer,” Opt. Commun. 425, 113–117 (2018).
[Crossref]

D. Wei and M. Aketagawa, “Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement,” Opt. Commun. 382, 604–609 (2017).
[Crossref]

D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Time-of-flight method using multiple pulse train interference as a time recorder,” Opt. Express 19(6), 4881–4889 (2011).
[Crossref] [PubMed]

D. Wei, S. Takahashi, K. Takamasu, and H. Matsumoto, “Analysis of the temporal coherence function of a femtosecond optical frequency comb,” Opt. Express 17(9), 7011–7018 (2009).
[Crossref] [PubMed]

Xia, X.-G.

X.-G. Xia, “Discrete chirp-Fourier transform and its application to chirp rate estimation,” IEEE Trans. Signal Process. 48(11), 3122–3133 (2000).
[Crossref]

Yamaguchi, I.

Yaroslavsky, L. P.

Zhang, F.

Bell Labs Tech. (1)

L. R. Rabiner, R. W. Schafer, and C. M. Rader, “The chirp z-transform algorithm and its application,” Bell Labs Tech. 48(5), 1249–1292 (1969).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

T. T. Wang, “The segmented chirp Z-transform and its application in spectrum analysis,” IEEE Trans. Instrum. Meas. 39(2), 318–323 (1990).
[Crossref]

IEEE Trans. Signal Process. (1)

X.-G. Xia, “Discrete chirp-Fourier transform and its application to chirp rate estimation,” IEEE Trans. Signal Process. 48(11), 3122–3133 (2000).
[Crossref]

J. Magn. Reson. (1)

J. Kaffanke, T. Dierkes, S. Romanzetti, M. Halse, J. Rioux, M. O. Leach, B. Balcom, and N. J. Shah, “Application of the chirp z-transform to MRI data,” J. Magn. Reson. 178(1), 121–128 (2006).
[Crossref] [PubMed]

Opt. Commun. (2)

D. Wei and M. Aketagawa, “Automatic selection of frequency domain filter for interference fringe analysis in pulse-train interferometer,” Opt. Commun. 425, 113–117 (2018).
[Crossref]

D. Wei and M. Aketagawa, “Time division approach to separate overlapped interference fringes of multiple pulse trains of femtosecond optical frequency comb for length measurement,” Opt. Commun. 382, 604–609 (2017).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Other (7)

M. Takeda, “Fourier Fringe Demodulation”, in Phase Estimation in Optical Interferometry (CRC Press, 2014), pp. 1–30.

S. A. Shilling, A study of the chirp Z-tranform and its applications, ((Unpublished doctoral dissertation), Kansas State University, Manhattan, KS., 1972).

B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom Algorithms for Digital Holography”, in Information Optics and Photonics: Algorithms, Systems, and Applications, B. Javidi and T. Fournel, eds. (Springer New York, New York, NY, 2010).

J. W. Goodman, Introduction to Fourier optics, 2nd ed., McGraw-Hill series in electrical and computer engineering (McGraw-Hill, New York, 1996).

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics (Cambridge University, 1996).

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (2nd ed.) (Prentice-Hall, Inc., 1999).

G. D. Martin, Chirp Z-transform spectral zoom optimization with MATLAB, (University of North Texas, 2019).

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Figures (2)

Fig. 1
Fig. 1 Configuration of the Michelson-type interferometer.
Fig. 2
Fig. 2 Fringe processing: (a) interference fringe signal (blue line), reconstructed whole envelope using DFT-based method (black dashed line) and partially reconstructed envelope using CZT-based method (red points). (b) The difference between the two obtained envelopes. The red downward arrow indicates the position of the envelope peak.

Equations (6)

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x( z n )x(n)=a( z n )+b( z n )| C t ( z n z eq ) |cos( k ¯ ( z n z eq )),
C DFT (n)= E DFT {x(n)}
E DFT {x(n)}2×abs{IDFT{BPF{DFT{x(n)}}}}
E DFT {x(n)}=2×abs{DFT{BPF{IDFT{x(n)}}}}
C CZT (N)= E CZT {x(n),N,w,a}
E CZT {x(n),N,w,a}=2×abs{CZT{BPF{IDFT{x(n)}},N,w,a}}

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