Abstract

Laser fringes have long been used to establish the x axis in interferometric spectrometry, but solutions for the intensity axis have been less satisfactory. Now we are seeing the rapid commercial development of low-cost, medium-speed, sigma–delta analog-to-digital converters developed for stereo audio applications. A single chip provides two channels of 20-bit precision at 50 kHz, a significant improvement over many current systems of much greater cost and complexity. But while the laser works in the spatial domain, this converter operates strictly in the time domain; it cannot be triggered. I have developed a bridge between these two domains, the adaptive digital filter, which not only permits us to use this converter to obtain measurements at arbitrary times but as a bonus shows us how to move much of the complexity of an interferometric-control and data-acquisition system from hardware to software. For example, flexible fringe subdivision (to increase the free spectral range) is easily obtained with a simple and efficient algorithm, completely free of laser ghosts. Compensation for drive velocity variation is also possible, requiring only a modest increase in computer memory.

© 1996 Optical Society of America

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References

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  1. J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the Fifteenth Advanced Course of the Swiss Society of Astronomy and Astrophysics, Saas-Fee, A. O. Benz, M. C. E. Huber, M. Mayor, eds. (Observatoire de Genève, Souverny, 1985), pp. 1–61.
  2. M. L. Forman, W. H. Steel, G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 56, 59–63 (1966).
    [CrossRef]

1966 (1)

Brault, J. W.

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the Fifteenth Advanced Course of the Swiss Society of Astronomy and Astrophysics, Saas-Fee, A. O. Benz, M. C. E. Huber, M. Mayor, eds. (Observatoire de Genève, Souverny, 1985), pp. 1–61.

Forman, M. L.

Steel, W. H.

Vanasse, G. A.

J. Opt. Soc. Am. (1)

Other (1)

J. W. Brault, “Fourier transform spectrometry,” in High Resolution in Astronomy, Proceedings of the Fifteenth Advanced Course of the Swiss Society of Astronomy and Astrophysics, Saas-Fee, A. O. Benz, M. C. E. Huber, M. Mayor, eds. (Observatoire de Genève, Souverny, 1985), pp. 1–61.

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Figures (8)

Fig. 1
Fig. 1

Measurement of fringe times with a simple interval timer.

Fig. 2
Fig. 2

Velocity variation in a relatively crude open-loop drive system. (a) Typical fringe times; note that the variation is mostly at low frequencies, with an rms variation of 2.2%. (b) Log amplitude spectrum of the fringe times in (a) relative to the mean time.

Fig. 3
Fig. 3

Using a cubic to interpolate between two fringe crossings.

Fig. 4
Fig. 4

Crude example of the formation of a set of interpolating DF’s with a 12-tap filter and only 4 divisions per sample.

Fig. 5
Fig. 5

(a) Response of a typical 50-tap filter for the second alias. (b) Top 4% of the response (scale on the right) and the relative sensitivity (defined as the ratio of ΔA/A to ΔV/V) to velocity variations (scale on the left). Attenuation is by roughly a factor of 50 in the central region, but falls rapidly to 1 when the response has fallen only a few percent.

Fig. 6
Fig. 6

Absolute value of the two sensitivity coefficients (dτ/τ0)/(dV/V) and (dA/A)/(dV/V) as a function of z = ω/ω0 for a single RC filter.

Fig. 7
Fig. 7

Diagram of the data acquisition and control system of the NOAA FTS.

Fig. 8
Fig. 8

(a) Monochromatic line observed without velocity-corrected sampling: The FWHM is ~7%, or 2650 points. (b) The same line observed with velocity-corrected sampling; the FWHM is now 3.4 points, correct for the Gaussian apodization used.

Equations (5)

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G ( t ) * DF ( 0 , - J ) G ( t ) * DF ( 1 , - J ) G ( t ) * DF ( m - 2 , - J ) G ( t ) * DF ( m - 1 , - J ) G ( t ) * DF ( 0 , - J + 1 ) G ( t ) * DF ( 1 , - J + 1 ) G ( t ) * DF ( m - 2 , - J + 1 ) G ( t ) * DF ( m - 1 , - J + 1 ) G ( t ) * DF ( 0 , J - 1 ) G ( t ) * DF ( 1 , J - 1 ) G ( t ) * DF ( m - 2 , J - 1 ) G ( t ) * DF ( m - 1 , J - 1 ) G ( t ) * DF ( 0 , + J ) G ( t ) * DF ( 1 , + J ) G ( t ) * DF ( m - 2 , + J ) G ( t ) * DF ( m - 1 , + J )
d A A = - z 2 1 + z 2 d V V
τ = ϕ ω = tan - 1 ( z ) ω ,             ω 0 τ = tan - 1 ( z ) z .
d τ τ 0 = [ 1 1 + z 2 - tan - 1 ( z ) z ] d V V = ( - 2 z 2 3 + 4 z 4 5 ) d V V .
g - 1 ( ω ) = 1 + j z ,

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