Abstract

We demonstrate a new technique for performing accurate Fourier transform interferometry with a 1-bit analog-to-digital (AD) converter that does not require oversampling of the interferogram, unlike in other 1-bit coding schemes that rely on delta-sigma modulation. Sampling aims at locating the intersections {z i} of the modulation term s(z) of the interferogram and a reference sinusoid r(z) = A cos(2πf r z), where z is the optical path difference. A new autocorrelation-based procedure that includes the accurate recovery of the equally sampled amplitude representation {s(k)} of s(z) from {z i} is utilized to calculate the square of the emission spectrum of the light source (sample). The procedure is suitable for interferograms that are corrupted with additive noise. Sinusoid-crossing sampling satisfies the Nyquist sampling criterion, and a z i exists within each sampling interval Δ = 1/2f r, if A ≥ |s(z)| for all z, and f rf c, where f c is the highest frequency component of s(z). By locating a crossing at an accuracy of 1 part in 216, we determine the multimode spectrum of an argon-ion laser with a 1-bit AD converter that performs like a 13-bit amplitude-sampling AD converter.

© 2000 Optical Society of America

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References

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  2. P. Hariharan, “Optical interferometry,” Rep. Prog. Phys. 54, 339–390 (1990).
    [CrossRef]
  3. J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).
  4. P. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley, New York, 1975).
  5. P. Grangier, J. Levenson, J. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
    [CrossRef]
  6. G. Hazel, F. Bucholtz, I. Aggarwal, “Characterization and modeling of drift noise in Fourier transform spectroscopy: implications for signal processing and detection limits,” Appl. Opt. 36, 6751–6759 (1993).
    [CrossRef]
  7. V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
    [CrossRef]
  8. M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
    [CrossRef]
  9. J. Proakis, D. Manolakis, Introduction to Digital Processing (Maxwell-Macmillan, 1989, New York), pp. 111–123.
  10. K. Minami, S. Kawata, “Dynamic range enhancement of Fourier transform infrared spectrum measurement using delta sigma modulation,” Appl. Opt. 32, 4822–4827 (1993).
    [CrossRef] [PubMed]
  11. C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
    [CrossRef]
  12. C. Saloma, P. Haeberli, “Optical spectrum analysis from zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
    [CrossRef] [PubMed]
  13. C. M. Blanca, V. Daria, C. Saloma, “Spectral recovery by analytic continuation in crossing-based spectral analysis,” Appl. Opt. 35, 6417–6422 (1996).
    [CrossRef] [PubMed]
  14. M. A. Nazario, C. Saloma, “Signal recovery in sinusoid-crossing sampling by use of the minimum-negativity constraint,” Appl. Opt. 37, 2953–2964 (1998).
    [CrossRef]
  15. M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
    [CrossRef]
  16. G. Pfeifer, “Modulators, demodulators and converters,” in Electronics Engineers Handbook, D. Fink, D. Christiansen, eds., (McGraw-Hill, New York, 1982), Section 14, pp. 14-24–14-45.
  17. M. Demler, High-Speed Analog-to-Digital Conversion (Academic, New York, 1991).
  18. J. Candy, “A use of double integration in delta signal modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
    [CrossRef]
  19. Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
    [CrossRef]
  20. K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Sys. 37, 309–318 (1990).
    [CrossRef]

1998 (4)

P. Grangier, J. Levenson, J. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
[CrossRef]

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

M. A. Nazario, C. Saloma, “Signal recovery in sinusoid-crossing sampling by use of the minimum-negativity constraint,” Appl. Opt. 37, 2953–2964 (1998).
[CrossRef]

1997 (1)

V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
[CrossRef]

1996 (1)

1993 (3)

1991 (1)

1990 (2)

P. Hariharan, “Optical interferometry,” Rep. Prog. Phys. 54, 339–390 (1990).
[CrossRef]

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Sys. 37, 309–318 (1990).
[CrossRef]

1987 (1)

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

1985 (1)

J. Candy, “A use of double integration in delta signal modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
[CrossRef]

Aggarwal, I.

Blanca, C. M.

Bucholtz, F.

Candy, J.

J. Candy, “A use of double integration in delta signal modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
[CrossRef]

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

Chao, K. C.

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Sys. 37, 309–318 (1990).
[CrossRef]

Daria, V.

V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
[CrossRef]

C. M. Blanca, V. Daria, C. Saloma, “Spectral recovery by analytic continuation in crossing-based spectral analysis,” Appl. Opt. 35, 6417–6422 (1996).
[CrossRef] [PubMed]

Demler, M.

M. Demler, High-Speed Analog-to-Digital Conversion (Academic, New York, 1991).

Grangier, P.

P. Grangier, J. Levenson, J. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
[CrossRef]

Griffiths, P.

P. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley, New York, 1975).

Haeberli, P.

Hariharan, P.

P. Hariharan, “Optical interferometry,” Rep. Prog. Phys. 54, 339–390 (1990).
[CrossRef]

Hazel, G.

Ishikawa, M.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

Iwata, A.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

Kawata, S.

Kobayashi, T.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

Lee, S. H.

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Sys. 37, 309–318 (1990).
[CrossRef]

Levenson, J.

P. Grangier, J. Levenson, J. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
[CrossRef]

Lim, M.

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

Litong, M.

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1975).

Manolakis, D.

J. Proakis, D. Manolakis, Introduction to Digital Processing (Maxwell-Macmillan, 1989, New York), pp. 111–123.

Matsuya, Y.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

Minami, K.

Nazario, M. A.

Pfeifer, G.

G. Pfeifer, “Modulators, demodulators and converters,” in Electronics Engineers Handbook, D. Fink, D. Christiansen, eds., (McGraw-Hill, New York, 1982), Section 14, pp. 14-24–14-45.

Poizat, J.

P. Grangier, J. Levenson, J. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
[CrossRef]

Proakis, J.

J. Proakis, D. Manolakis, Introduction to Digital Processing (Maxwell-Macmillan, 1989, New York), pp. 111–123.

Saloma, C.

M. A. Nazario, C. Saloma, “Signal recovery in sinusoid-crossing sampling by use of the minimum-negativity constraint,” Appl. Opt. 37, 2953–2964 (1998).
[CrossRef]

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
[CrossRef]

C. M. Blanca, V. Daria, C. Saloma, “Spectral recovery by analytic continuation in crossing-based spectral analysis,” Appl. Opt. 35, 6417–6422 (1996).
[CrossRef] [PubMed]

C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
[CrossRef]

C. Saloma, P. Haeberli, “Optical spectrum analysis from zero crossings,” Opt. Lett. 16, 1535–1537 (1991).
[CrossRef] [PubMed]

Sodini, C. G.

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Sys. 37, 309–318 (1990).
[CrossRef]

Uchimura, K.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

Yoshitome, T.

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

Appl. Opt. (4)

IEEE J. Solid-State Circuits (1)

Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, M. Ishikawa, T. Yoshitome, “A 16-bit oversampling A-to-D conversion technology using triple integration noise shaping,” IEEE J. Solid-State Circuits SC-22, 921–929 (1987).
[CrossRef]

IEEE Trans. Circuits Sys. (1)

K. C. Chao, S. H. Lee, C. G. Sodini, “A high order topology for interpolative modulators for oversampling A/D converter,” IEEE Trans. Circuits Sys. 37, 309–318 (1990).
[CrossRef]

IEEE Trans. Commun. (1)

J. Candy, “A use of double integration in delta signal modulation,” IEEE Trans. Commun. COM-33, 249–258 (1985).
[CrossRef]

J. Appl. Phys. (1)

C. Saloma, “Computational complexity and observation of physical signals,” J. Appl. Phys. 74, 5314–5319 (1993).
[CrossRef]

Nature (1)

P. Grangier, J. Levenson, J. Poizat, “Quantum non-demolition measurements in optics,” Nature 396, 537–542 (1998).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (2)

M. Lim, C. Saloma, “Direct signal recovery from threshold crossings,” Phys. Rev. E 58, 6759–6765 (1998).
[CrossRef]

M. Litong, C. Saloma, “Detection of sub-threshold oscillations by sinusoid-crossing sampling,” Phys. Rev. E 57, 3579–3588 (1998).
[CrossRef]

Rep. Prog. Phys. (1)

P. Hariharan, “Optical interferometry,” Rep. Prog. Phys. 54, 339–390 (1990).
[CrossRef]

Rev. Sci. Instrum. (1)

V. Daria, C. Saloma, “Bandwidth and detection limit in crossing-based spectrum analyzer,” Rev. Sci. Instrum. 68, 240–242 (1997).
[CrossRef]

Other (6)

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley, New York, 1979).

P. Griffiths, Chemical Infrared Fourier Transform Spectroscopy (Wiley, New York, 1975).

G. Pfeifer, “Modulators, demodulators and converters,” in Electronics Engineers Handbook, D. Fink, D. Christiansen, eds., (McGraw-Hill, New York, 1982), Section 14, pp. 14-24–14-45.

M. Demler, High-Speed Analog-to-Digital Conversion (Academic, New York, 1991).

J. Proakis, D. Manolakis, Introduction to Digital Processing (Maxwell-Macmillan, 1989, New York), pp. 111–123.

D. Malacara, Optical Shop Testing (Wiley, New York, 1975).

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Figures (5)

Fig. 1
Fig. 1

Procedure for calculating the power spectrum |S(f)|2 of s(z) from its equally sampled representation {s(k)}, which is computed from {z i } by use of the ideal interpolation formula. The spectrum |S(f)|2 is the discrete FT of the autocorrelation γ(k) of {s(k)}. In FT spectroscopy, s(z) represents the interferogram, and the emission (absorption) spectrum of the light source (sample) is given by S(f) = ℱ[{s(k)}].

Fig. 2
Fig. 2

Delta-sigma modulation. NMSE of the sampled amplitude representation of s(z) = Asin(2πf c z), where A = 1, 0 ≤ f c ≤ 120 Hz, and 1/Δ = 65536 Hz. The following values were considered for the single-integrator delta-sigma modulation: Δs equal to 1/28, 1/29, and 1/210.

Fig. 3
Fig. 3

Experimental setup of the FT spectrometer. Mirror M1 is scanned relative to M2 by use of a piezoelectric transducer that is driven with a 0.5-Hz triangular wave. The interference pattern produced by the Michelson interferometer is detected by a phototransistor. The 1-bit AD converter consists of a single comparator and a FPGA.

Fig. 4
Fig. 4

Calibration of the FT spectrometer. Plots of wave number σ(cm-1) versus the Fourier coefficient index number n for three He–Ne lasers emitting at σ = 99291, 105759, and 115606 cm-1. The solid line (best-fit curve) is described by σ(cm-1) = 167.78n - 12994.

Fig. 5
Fig. 5

Power spectrum |S(σ)|2 of a multimode argon-ion laser as measured by the FT spectrometer. Also shown are the spectral curves of the He–Ne laser that were used in the calibration. The corresponding emission spectrum is given by |S(σ)| = (|S(σ)|2)1/2.

Equations (16)

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

δ=Δ/2q=T/M2q+1,
zi=i-12qδ+piδ,
szi=rzi±Ezi,
Ezi=1/2|rz/zdz|,
Ezi=Aπ/2q+1|sinπzi/2qδ|.
EziAπ2pi/22q+2,
EziAπ/2q+1πzi/2qδ,
Ezi=EmaxAπ/2q+1.
2q-1fs=2qfr=1/δc.
Cf=sk+nk
|Cf|2=γsk+γnk+γsnk+γnsk,
|Cf|2γsk+γnk.
Iz=κ+sz
rt=A cos2πfrt,
σcm-1=167.78n-12994,
σcm-1=167.78n-12994.

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