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A “virtual-interferometer” technique for surface metrology

Scott M. Jobling and Paul G. Kwiat

Open Access Open Access

Abstract

We demonstrate a novel technique for performing aberration-corrected surface metrology within existing wavefront-feedback systems. Our technique uses several phase measurements to calculate phase differences that directly reveal the surface gradients of an object under test, due to orthogonal displacements of that object between measurements. We then apply a least-squares algorithm for surface reconstruction using the gradient information. This approach also removes static system aberrations, providing an absolute measurement of the surface profile. To date, we have profiled a number of test optics with 20- to 40-nm RMS error, where the accuracy is determined by the amount of angular crosstalk over the system aperture.

©2010 Optical Society of America

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Figures (7)
Fig. 1
Fig. 1 An interferometric approach for surface metrology. (a) Experimental layout. (b) Example fringe pattern measured by camera.

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Fig. 2
Fig. 2 A virtual-interferometric approach for surface metrology. (a) Experimental layout, in which the wavefront from each arm is measured separately, and the results are subtracted to determine the surface shape of the object under test. (b) Example wavefront difference calculated from individually measured wavefronts.

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Fig. 3
Fig. 3 A single-arm virtual-interferometric approach for surface metrology: F1 and F2 are the focal lengths of lenses arranged in a 4f-imaging configuration, used to image the surface of the object under test onto the wavefront sensor.

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Fig. 4
Fig. 4 Displaced-object wavefront measurements. (a) Initial position. (b) Displaced horizontally. (c) Difference between (a) and (b), which corresponds to the gradient of a spherical surface along the direction of displacement.

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Fig. 5
Fig. 5 Experimental gradient reconstruction data for a deformable mirror approximating a trefoil shape. (a) Horizontal gradient. (b) Vertical gradient. (c) Reconstructed surface.

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Fig. 6
Fig. 6 Surface data for various optical elements. Each measurement used the 200:100-mm imaging system, with a maximum aperture of 12x9.6 mm2. The numbers on the plots indicate the measured / specified parameters for each element (X indicating unknown). (a-c) Flat mirror profiles and RMS-flatness data in nanometers. Note: specified value is a maximum tolerance. (d-e) Spherical mirror profiles and radius of curvature data in meters.

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Fig. 7
Fig. 7 Demonstration of background removal with the SAVI technique. (a) Flat mirror reconstructed with unaberrated beam path. (b) Measured aberrations added to beam path by a microscope slide. (c) Flat mirror reconstructed with aberrated beam path.

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Equations (6)

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( C ) = ( ∇ x ϕ ∇ y ϕ ) × ( A ) , and
( ∇ x ϕ ' ∇ y ϕ ' ) = ( ∇ x ϕ ∇ y ϕ ) − ( C ) × ( A T ) , where
( A ) = ( ∇ x ϕ T i l t 0 ∘ ∇ y ϕ T i l t 0 ∘ ∇ x ϕ T i l t 90 ∘ ∇ y ϕ T i l t 90 ∘ ∇ x ϕ A s t i g 0 ∘ ∇ y ϕ A s t i g 0 ∘ ∇ x ϕ A s t i g 45 ∘ ∇ y ϕ A s t i g 45 ∘ ∇ x ϕ F o c ∇ y ϕ F o c ) , and
ϕ j , k i = ϕ ¯ j , k i + w j , k b j , k i , and
ϕ ¯ j , k i = w ϕ ¯ j k ( ϕ j + 1 , k i + ϕ j , k + 1 i + ϕ j − 1 , k i + ϕ j , k − 1 i ) ,
b j , k i = w b j k ( ∇ y ϕ j − 1 , k i − ∇ y ϕ j , k i + ∇ x ϕ j , k − 1 i − ∇ x ϕ j , k i ) .
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