Abstract

A sensitive interferometric sensor scheme that is based on coherent imaging of a first phase grating onto a second phase grating, their periods accurately matched, is suggested. Experimental data, obtained with a setup based on the suggested scheme, are presented. The sensor was found capable of measuring an angular tilt of a mirror less than 0.5 µrad. Compared with a previously suggested measuring scheme, the novelty of the one presented here is the inclusion of a second set of gratings, which eliminates measurement ambiguity. Some characteristics of the sensor scheme are discussed.

© 2003 Optical Society of America

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References

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  1. D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Material (Springer-Verlag, New York, 1994).
  2. W. Schmann, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1985).
  3. A. Magnusson, S. Hård, “Four-level phase grating generated by combination of two gratings through coherent imaging,” Appl. Opt. 39, 5936–5939 (2000).
    [CrossRef]
  4. W. A. Nash, Schaum’s Outline of Theory and Problems of Strength of Materials, 4th ed. (McGraw-Hill, New York, 1998).
  5. C. Narayanan, A. B. Buckman, I. Busch-Vishniac, “Noise analysis for position-sensitive detectors,” IEEE Trans. Instrum. Meas. 46, 1137–1144 (1997).
    [CrossRef]

2000 (1)

1997 (1)

C. Narayanan, A. B. Buckman, I. Busch-Vishniac, “Noise analysis for position-sensitive detectors,” IEEE Trans. Instrum. Meas. 46, 1137–1144 (1997).
[CrossRef]

Buckman, A. B.

C. Narayanan, A. B. Buckman, I. Busch-Vishniac, “Noise analysis for position-sensitive detectors,” IEEE Trans. Instrum. Meas. 46, 1137–1144 (1997).
[CrossRef]

Busch-Vishniac, I.

C. Narayanan, A. B. Buckman, I. Busch-Vishniac, “Noise analysis for position-sensitive detectors,” IEEE Trans. Instrum. Meas. 46, 1137–1144 (1997).
[CrossRef]

Han, B.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Material (Springer-Verlag, New York, 1994).

Hård, S.

Ifju, P.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Material (Springer-Verlag, New York, 1994).

Magnusson, A.

Narayanan, C.

C. Narayanan, A. B. Buckman, I. Busch-Vishniac, “Noise analysis for position-sensitive detectors,” IEEE Trans. Instrum. Meas. 46, 1137–1144 (1997).
[CrossRef]

Nash, W. A.

W. A. Nash, Schaum’s Outline of Theory and Problems of Strength of Materials, 4th ed. (McGraw-Hill, New York, 1998).

Post, D.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Material (Springer-Verlag, New York, 1994).

Schmann, W.

W. Schmann, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1985).

Appl. Opt. (1)

IEEE Trans. Instrum. Meas. (1)

C. Narayanan, A. B. Buckman, I. Busch-Vishniac, “Noise analysis for position-sensitive detectors,” IEEE Trans. Instrum. Meas. 46, 1137–1144 (1997).
[CrossRef]

Other (3)

W. A. Nash, Schaum’s Outline of Theory and Problems of Strength of Materials, 4th ed. (McGraw-Hill, New York, 1998).

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Material (Springer-Verlag, New York, 1994).

W. Schmann, Holography and Deformation Analysis (Springer-Verlag, Berlin, 1985).

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Figures (4)

Fig. 1
Fig. 1

Schematic of a 4f-imaging, transmission-mode, interferometric laser sensor system. G1 and G2 are 50% duty cycle, binary phase gratings. The period of G1 is twice that of G2. The grating phase steps of G1 and G2 are 180° and 90°, respectively. A phase object PO to be measured is positioned in one of the two beams diffracted by G1 (B2); PO is located halfway between the two lenses L1 and L2. D1 and D2 are photodetectors.

Fig. 2
Fig. 2

Schematic of the reflection-mode, experimental setup. BE, horizontal beam expander; BS1 and BS2, beam splitters, and Apt, two-hole aperture. G1 and G1′, linear binary phase gratings, which have horizontal grooves with 12.0-µm period and 180° phase step; G1 is shifted vertically 1.5 µm relative to G1′. L, lens, with a focal length of f = 90 mm, that images G1 and G1′ onto G2 and G2′, respectively. M, mirror, whose angular change Δφ is measured, that has a horizontal axis of rotation. G2 and G2′, linear binary phase gratings that have horizontal grooves with 6.0-µm period and 90° phase step. D and D′, photodiodes, with current-to-voltage converters built in.

Fig. 3
Fig. 3

Grating arrangement on the quartz substrate in the experimental setup shown in Fig. 2. The cross indicates the position of the optical axis. G1 and G1′ have the grating period Λ1 = 12.0 µm and a 180° phase step. The period of G2 and G2′ is Λ2 = 6.0 µm, and they have a phase step of 90°. The grooves of G1 and G1′ are internally shifted vertically Λ2/4 = 1.5 µm, and those of G2 and G2′ are unshifted. The duty cycle of all four gratings is 50%.

Fig. 4
Fig. 4

Oscilloscope traces from the two detectors obtained by repeatedly loading and unloading a weight of 8.1 g at the center of an aluminum beam, with a mirror attached to its end. The top trace shows the signal from the first detector, operating at maximum angular sensitivity, and the lower trace shows the signal from the detector receiving minimum power.

Equations (8)

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P1δ=K1+0.906 cos2πδ/λ0+θ,
P2δ=K1-0.906 cos2πδ/λ0+θ,
P1δ=K1+0.906 sin2πδ/λ0+θ,
P2δ=K1-0.906 sin2πδ/λ0+θ,
Sδ=cos2πδ/λ0+θ,
Sδ=sin2πδ/λ0+θ,
Δφ=FL216EIs,
Δδ/B1/20.25λ0eIo1/2=0.25hcλ0Pη1/2,

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