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Optical tests with Bessel beam interferometry

Mathieu Fortin, Michel Piché, and Ermanno F. Borra

Open Access Open Access

Abstract

In this paper we demonstrate how Bessel beam interferometry can be used to characterize the curvature of a reflecting surface. The approach is based on the fact that the intensity distribution produced by the coherent superposition of Bessel beams is a sensitive function of the relative phases between the constituting beams. We show how this phase sensitivity can translate into accurate measurements of the curvature of a wavefront. Experimental tests were made with a liquid mirror. We have also used Bessel beams to measure the precession angle of the liquid mirror. Our results show that Bessel beam interferometry is a very accurate tool for the optical testing of non-stationary surfaces and that it could be used as a general method of real-time, non-contact sensing. Bessel beam interferometry has the advantage of not requiring any reference arm that needs to be stabilized.

©2004 Optical Society of America

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Figures (7)
Fig 1.
Fig 1. (a) Single Bessel beam generated by using a mask with one thin annular slit. (b) Constructive interference of two Bessel beams produced by using a mask with two concentric annular slits. (c) Destructive interference of two Bessel beams produced by using a mask with two concentric annular slits. (d) Off-axis Bessel beam generated by an incident beam with a tilted wavefront.

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Fig. 2.
Fig. 2. Coherent superposition of two Bessel beams. Upper left: intensity at the center of the interference pattern produced with a mask having two annular slits, as a function of the curvature of the beam incident on the mask. Lower left: interference pattern for the three values of curvature indicated above by points i, ii and iii. Upper right: intensity at the center of the interference pattern produced with a mask having one annular slit and one hole at center. Lower right: interference pattern for the three values of curvature indicated above by points iv, v and vi.

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Fig. 3.
Fig. 3. Experimental setup used for the characterization of the curvature of a liquid mirror based on Bessel beam interferometry.

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Fig. 4.
Fig. 4. Variation of the intensity at the center of the Bessel beam interference pattern as a function of the radius of curvature of the liquid mirror. Experimental data points (crosses) were obtained with the mask having two annular slits. The continuous curve is the theoretical prediction. Measurements were made in varying the value of Rm by steps of 200 µm.

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Fig. 5.
Fig. 5. Interference patterns observed with a two-slit mask, for values of the radius of curvature of the liquid mirror Rm between 1.9996 m and 2.0167 m.

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Fig. 6.
Fig. 6. Cardioid motion of the center of curvature of the liquid mirror during one revolution. Eight measurements were made, by angular steps of 45 degrees.

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Fig. 7.
Fig. 7. Interference patterns produced by the superposition of a Bessel beam and an Airy pattern, with R1=15 cm, for different phases during a revolution of the liquid mirror.

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Equations (2)

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

E tot ( r , z = 2 f ) ≈ ∑ n = 1 N A n exp ( − i k d n 2 4 R 1 ) J 0 ( α n r )
I tot ( r = 0 , z = 2 f ) ≈ B 1 2 + B 2 2 + 2 B 1 B 2 cos ( k ( d 2 2 − d 1 2 ) 8 R 1 )
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