## 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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### Equations (5)

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(1)
$$\begin{array}{ccccc}\text{G}(\text{t})*\text{DF}(0,-\text{J})& \text{G}(\text{t})*\text{DF}(1,-\text{J})& \dots & \text{G}(\text{t})*\text{DF}(\text{m}-2,-\text{J})& \text{G}(\text{t})*\text{DF}(\text{m}-1,-\text{J})\\ \text{G}(\text{t})*\text{DF}(0,-\text{J}+1)& \text{G}(\text{t})*\text{DF}(1,-\text{J}+1)& \dots & \text{G}(\text{t})*\text{DF}(\text{m}-2,-\text{J}+1)& \text{G}(\text{t})*\text{DF}(\text{m}-1,-\text{J}+1)\\ \dots & \dots & \dots & \dots & \dots \\ \text{G}(\text{t})*\text{DF}(0,\text{J}-1)& \text{G}(\text{t})*\text{DF}(1,\text{J}-1)& \dots & \text{G}(\text{t})*\text{DF}(\text{m}-2,\text{J}-1)& \text{G}(\text{t})*\text{DF}(\text{m}-1,\text{J}-1)\\ \text{G}(\text{t})*\text{DF}(0,+\text{J})& \text{G}(\text{t})*\text{DF}(1,+\text{J})& \dots & \text{G}(\text{t})*\text{DF}(\text{m}-2,+\text{J})& \text{G}(\text{t})*\text{DF}(\text{m}-1,+\text{J})\end{array}$$
(2)
$$\frac{\text{d}A}{A}=-\frac{{z}^{2}}{1+{z}^{2}}\frac{\text{d}V}{V}$$
(3)
$$\mathrm{\tau}=\frac{\mathrm{\varphi}}{\mathrm{\omega}}=\frac{{\text{tan}}^{-1}(z)}{\mathrm{\omega}},\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{\mathrm{\omega}}_{0}\mathrm{\tau}=\frac{{\text{tan}}^{-1}(z)}{z}.$$
(4)
$$\frac{\text{d}\mathrm{\tau}}{{\mathrm{\tau}}_{0}}=\left[\frac{1}{1+{z}^{2}}-\frac{{\text{tan}}^{-1}(z)}{z}\right]\frac{\text{d}V}{V}=\left(-\frac{2{z}^{2}}{3}+\frac{4{z}^{4}}{5}\cdots \right)\frac{\text{d}V}{V}.$$
(5)
$${g}^{-1}(\mathrm{\omega})=1+jz,$$