Abstract

Surface measurements of precision optics are commonly made with commercially available phase-shifting Fizeau interferometers that provide data relative to flat or spherical reference surfaces whose unknown errors are comparable to those of the surface being tested. A number of ingenious techniques provide surface measurements that are “absolute,” rather than relative to any reference surface. Generally, these techniques require numerous measurements and the introduction of additional surfaces, but still yield absolute information only along certain lines over the surface of interest. A very simple alternative is presented here, in which no additional optics are required beyond the surface under test and the transmission flat (or sphere) defining the interferometric reference surface. The optic under test is measured in three positions, two of which have small lateral shifts along orthogonal directions, nominally comparable to the transverse spatial resolution of the interferometer. The phase structure in the reference surface then cancels out when these measurements are subtracted in pairs, providing a grid of absolute surface height differences between neighboring resolution elements of the surface under test. The full absolute surface, apart from overall phase and tip/tilt, is then recovered by standard wavefront reconstruction techniques.

© 2010 Optical Society of America

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References

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2008 (1)

J. C. Marr IV, M. Shao, and R. Goullioud, Proc. SPIE 7013, 70132M (2008).
[CrossRef]

2006 (1)

2001 (1)

1994 (1)

1993 (1)

1980 (1)

1977 (1)

Ai, C.

Freischlad, K. R.

Goullioud, R.

J. C. Marr IV, M. Shao, and R. Goullioud, Proc. SPIE 7013, 70132M (2008).
[CrossRef]

Griesmann, U.

Grzanna, J.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

Herrmann, J.

Hudgin, R. H.

Marr, J. C.

J. C. Marr IV, M. Shao, and R. Goullioud, Proc. SPIE 7013, 70132M (2008).
[CrossRef]

Schulz, G.

G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E.Wolf, ed. (Elsevier, 1976), p. 93–167.
[CrossRef]

Schwider, J.

G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E.Wolf, ed. (Elsevier, 1976), p. 93–167.
[CrossRef]

Shao, M.

J. C. Marr IV, M. Shao, and R. Goullioud, Proc. SPIE 7013, 70132M (2008).
[CrossRef]

Wyant, J. C.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Proc. SPIE (1)

J. C. Marr IV, M. Shao, and R. Goullioud, Proc. SPIE 7013, 70132M (2008).
[CrossRef]

Other (2)

G. Schulz and J. Schwider, in Vol. 13 of Progress in Optics, E.Wolf, ed. (Elsevier, 1976), p. 93–167.
[CrossRef]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998).

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Figures (3)

Fig. 1
Fig. 1

Measurement schematic: the surface height of the mirror under test is mapped at subnanometer precision, but only relative to the transmission flat (reference surface), typically unknown and flat to just a few nanometers.

Fig. 2
Fig. 2

Phase maps measured by the interferometer of Fig. 1, with slight transverse shifts along two axes of the mirror with respect to the transmission flat that provides the reference surface. In each of the three measurements, the interferometer camera is fixed with respect to the reference surface (transmission flat or sphere).

Fig. 3
Fig. 3

Simulation of wavefront reconstruction on absolute difference data from a single Fizeau map: (a) 70 × 70 map, here modeling the absolute test surface before corruption by the reference surface (transmission flat). Color bars are in nanometers. (b) Map of differences of Y (vertical)-neighbor pixels for the previous map. These absolute differences, along with those in the X direction, may be measured free of transmission-flat corruption using the spatial differencing procedure described in the text. (c) Reconstructed absolute test surface using Eq. (6); the accurate recovery of the absolute test surface of frame (a) illustrates the proposed technique.

Equations (7)

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Φ 0 ( x , y ) = ϕ mirror ( x , y ) ψ ref ( x , y ) ,
Φ Δ x ( x , y ) = ϕ mirror ( x + Δ x , y ) ψ ref ( x , y ) , Φ Δ y ( x , y ) = ϕ mirror ( x , y + Δ y ) ψ ref ( x , y ) .
Φ Δ x Φ 0 = ϕ mirror ( x + Δ x , y ) ϕ mirror ( x , y ) , Φ Δ y Φ 0 = ϕ mirror ( x , y + Δ y ) ϕ mirror ( x , y ) .
S = A W + n ,
A [ q + ( p 1 ) ( N 1 ) , q + ( p 1 ) N ] = 1 , A [ q + ( p 1 ) ( N 1 ) , 1 + q + ( p 1 ) N ] = 1 for p = 1 , , N , q = 1 , , N 1 ; A [ r + N ( N 1 ) , r ] = 1 , A [ r + N ( N 1 ) , r + N ] = 1 for r = 1 , , N ( N 1 ) .
W = ( A T A ) 1 A T S .
σ 2 wf = E σ 2 in = [ a + b ln N 2 ] σ 2 in ,

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