Abstract

We have demonstrated several inexpensive methods that can be used to measure the deflection angles of prisms with microradian precision. The methods are self-referenced, where various reversals are used to achieve absolute measurements without the need of a reference prism or any expensive precision components other than the prisms under test. These techniques are based on laser interferometry and have been used in our laboratory to characterize parallel-plate beam splitters, penta prisms, right-angle prisms, and corner cube reflectors using only components typically available in an optics laboratory.

© 2005 Optical Society of America

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References

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  1. J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
    [CrossRef] [PubMed]
  2. M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 1–50.
  3. D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 51–94.
  4. Z. Malacara, “Angle, distance, curvature, and focal length,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 715–742.
  5. M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 123–172.
  6. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501–598.

1994 (1)

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Bender, P. L.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Bruning, J. H.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501–598.

Dickey, J. O.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Faller, J. E.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Greivenkamp, J. E.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501–598.

Malacara, D.

D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 51–94.

Malacara, Z.

Z. Malacara, “Angle, distance, curvature, and focal length,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 715–742.

Mantravadi, M. V.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 123–172.

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 1–50.

Newhall, X. X.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Ricklefs, R. L.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Ries, J. G.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Shelus, P. J.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Veillet, C.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Whipple, A. L.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Wiant, J. R.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Williams, J. G.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Yoder, C. F.

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Science (1)

J. O. Dickey, P. L. Bender, J. E. Faller, X. X. Newhall, R. L. Ricklefs, J. G. Ries, P. J. Shelus, C. Veillet, A. L. Whipple, J. R. Wiant, J. G. Williams, C. F. Yoder, “Lunar laser ranging: a continuing legacy of the Apollo program,” Science 265, 482–490 (1994).
[CrossRef] [PubMed]

Other (5)

M. V. Mantravadi, “Newton, Fizeau, and Haidinger interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 1–50.

D. Malacara, “Twyman-Green interferometer,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 51–94.

Z. Malacara, “Angle, distance, curvature, and focal length,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 715–742.

M. V. Mantravadi, “Lateral shearing interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 123–172.

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometers,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, 1992), pp. 501–598.

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Figures (9)

Fig. 1
Fig. 1

Fringes formed by two crossing plane waves. In (a) the interference pattern formed when two beams from a He–Ne laser crossed at a small angle is shown. In (b) the gray arrows represent the propagation or k vectors of the two plane waves, and the thin gray lines represent the wave fronts of the two traveling waves. The angle between the propagation vectors of the two beams is labeled as Δθ. The interference maxima, where the two waves are always in phase, are denoted with the dashed lines, and the spacing between the interference maxima is labeled as d.

Fig. 2
Fig. 2

Calculation of the normalized interference pattern. Images (a)–(d) are an example of the four images that are needed to evaluate Eq. (4). Frames (a) and (b) are images of the individual beams with the other beam blocked, (c) is a dark field with both beams blocked, and (d) is an image of the two beams interfering. The closely spaced interference lines visible in these images are low-contrast fringes due to reflections off of the camera window and the focusing lens. The high-contrast fringes due to the angle between the two beams are not apparent in the interference frame because the spacing between fringes is larger than the size of the beams. Plugging the data from these images into the left-hand side of Eq. (4) results in the image shown in (e). Only the central part of (e), where both beams are present, contains meaningful information. The shading scale in (e) runs from −1.35 (pure black) to 0.05 (pure white).

Fig. 3
Fig. 3

Curve fits to find the angle between two beams. Strips through the center of the data from Fig. 2(e) are shown, along with least-squared fits to the functions cos(kxx + ϕx) and cos(kyy + ϕy). The deviation of the data from the fits is largely due to camera window reflections. These higher-spatial-frequency, low-contrast fringes average away to a large extent in the curve fit.

Fig. 4
Fig. 4

Generation of two nearly parallel beams with a plate beam splitter. The gray lines represent laser light. Light enters the beam splitter in the lower left-hand corner. At each interface the beam is split into a reflected and a transmitted beam. For most of our studies we are interested only in the two beams exiting the beam splitter that are labeled 1 and 2. The angle between the incoming beam and the normal of the first surface is labeled as γ, the angle between beams 1 and 2 is labeled as θ, and the wedge angle of the glass plate is labeled ψ.

Fig. 5
Fig. 5

Optical setup to measure the wedge angles of parallel-plate beam splitters.

Fig. 6
Fig. 6

Optical setup to measure relative deflection angles of penta prisms.

Fig. 7
Fig. 7

Finding the relative deflection error of two penta prisms. The magnitude of the wave vector describing the interference pattern at different prism alignments is plotted versus (a) the PZT voltage and (b) the y component of the wave vector. The crosses and the asterisks represent the actual data extracted from the interference patterns. The asterisks represent the data points that should be the most accurate since the image happened to fall between a light and a dark fringe. The crosses represent data points for which the image contained a light or dark extremum. The curves represent equally weighted least-squares fits of the entire data set to Eq. (10). Data from a different set of prisms that did not meet our specifications are shown in (c).

Fig. 8
Fig. 8

Optical setup used to measure the relative deflection of two right-angle prisms or corner cubes.

Fig. 9
Fig. 9

Optical setup for absolute measurement of right-angle prism and corner cube deflection angles.

Equations (10)

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Δ θ = λ / d .
E n ( r , t ) = f n ( r ) cos ( k n · r - ω t + ϕ n ) ,
I 12 ( r ) = I 1 + I 2 + 2 ( I 1 I 2 ) 1 / 2 cos ( k rel · r + Δ ϕ ) ,
I 12 - I 1 - I 2 2 ( I 1 I 2 ) 1 / 2 = cos ( k x x + k y y + Δ ϕ ) .
ψ x = θ x 2 [ 1 - sin 2 ( γ ) n 2 - sin 2 ( γ ) ] 1 / 2 ,
M 1 x = θ A x - θ B x ,
± M 2 x = θ A x + θ B x ,
± M 3 x = θ A x - θ C x ,
± M 4 x = θ A x + θ C x .
k rel = ( k p 2 + k a 2 ) 1 / 2 .

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