Abstract

The concept of a differential Shack–Hartmann (DSH) curvature sensor was recently proposed, which yields wavefront curvatures by measuring wavefront slope differentials. As an important feature of the DSH curvature sensor, the wavefront twist curvature terms can be efficiently obtained from slope differential measurements, thus providing a means to measure the Monge-equivalent patch. Specifically, the principal curvatures and principal directions, four key parameters in differential geometry, can be computed from the wavefront Laplacian and twist curvature terms. The principal curvatures and directions provide a “complete” definition of wavefront local shape. Given adequate sampling, these measurements can be useful in quantifying the mid-spatial-frequency wavefront errors, yielding a complete characterization of the surface being measured.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  4. I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
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2005 (2)

W. Zou and J. Rolland, “Differential wavefront curvature sensor,” Proc. SPIE 5869, 5869171 (2005).

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592-614 (2005).
[CrossRef]

2002 (1)

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156-1166 (2002).
[CrossRef]

2001 (2)

M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttgart) 112, 86-90 (2001).
[CrossRef]

I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
[CrossRef]

2000 (1)

1999 (1)

I. Weingaertner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306-317 (1999).
[CrossRef]

1997 (1)

V. Interrante, H. Fuchs, and S. M. Pizer, “Conveying the 3D shape of smoothly curving transparent surface via texture,” IEEE Trans. Vis. Comput. Graph. 3, 98-117 (1997).
[CrossRef]

1996 (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

1993 (1)

1992 (1)

1990 (3)

P. E. Glenn, “Robust, sub-angstrom level mid-spatial frequency profilometry,” Proc. SPIE 1333, 230-238 (1990).
[CrossRef]

P. Glenn, “Angstrom level profilometry for sub-millimeter to meter scale surface errors,” Proc. SPIE 1333, 326-336 (1990).
[CrossRef]

M. Sarazin and F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294-300 (1990).

1988 (1)

1971 (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Dainty, J. C.

Elster, C.

I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
[CrossRef]

I. Weingaertner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306-317 (1999).
[CrossRef]

Fuchs, H.

V. Interrante, H. Fuchs, and S. M. Pizer, “Conveying the 3D shape of smoothly curving transparent surface via texture,” IEEE Trans. Vis. Comput. Graph. 3, 98-117 (1997).
[CrossRef]

Glenn, P.

P. Glenn, “Angstrom level profilometry for sub-millimeter to meter scale surface errors,” Proc. SPIE 1333, 326-336 (1990).
[CrossRef]

Glenn, P. E.

P. E. Glenn, “Robust, sub-angstrom level mid-spatial frequency profilometry,” Proc. SPIE 1333, 230-238 (1990).
[CrossRef]

Guyon, O.

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592-614 (2005).
[CrossRef]

Hardy, J.

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

Interrante, V.

V. Interrante, H. Fuchs, and S. M. Pizer, “Conveying the 3D shape of smoothly curving transparent surface via texture,” IEEE Trans. Vis. Comput. Graph. 3, 98-117 (1997).
[CrossRef]

Koenderink, J. J.

J. J. Koenderink, Solid Shape (MIT, 1990), pp. 210, 214, 212, 228, 232.

Paterson, C.

Pizer, S. M.

V. Interrante, H. Fuchs, and S. M. Pizer, “Conveying the 3D shape of smoothly curving transparent surface via texture,” IEEE Trans. Vis. Comput. Graph. 3, 98-117 (1997).
[CrossRef]

Platt, B. C.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Ragazzoni, R.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

Roddier, C.

Roddier, F.

Rolland, J.

W. Zou and J. Rolland, “Differential wavefront curvature sensor,” Proc. SPIE 5869, 5869171 (2005).

W. Zou and J. Rolland, “Differential Shack-Hartmann curvature sensor,” U.S. patent 7,390,999 (24 June 2008).

Sarazin, M.

M. Sarazin and F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294-300 (1990).

Schulz, M.

I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
[CrossRef]

M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttgart) 112, 86-90 (2001).
[CrossRef]

I. Weingaertner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306-317 (1999).
[CrossRef]

Shack, R. V.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Thomsen-Schmidt, P.

I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
[CrossRef]

Tippur, H. V.

Tokovinin, A.

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156-1166 (2002).
[CrossRef]

Weingaertner, I.

I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
[CrossRef]

I. Weingaertner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306-317 (1999).
[CrossRef]

Zou, W.

W. Zou and J. Rolland, “Differential wavefront curvature sensor,” Proc. SPIE 5869, 5869171 (2005).

W. Zou, “Optimization of zonal wavefront estimation and curvature measurements,” Ph.D. dissertation (University of Central Florida, 2007).

W. Zou and J. Rolland, “Differential Shack-Hartmann curvature sensor,” U.S. patent 7,390,999 (24 June 2008).

Appl. Opt. (2)

Astron. Astrophys. (1)

M. Sarazin and F. Roddier, “The ESO differential image motion monitor,” Astron. Astrophys. 227, 294-300 (1990).

Astrophys. J. (1)

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592-614 (2005).
[CrossRef]

IEEE Trans. Vis. Comput. Graph. (1)

V. Interrante, H. Fuchs, and S. M. Pizer, “Conveying the 3D shape of smoothly curving transparent surface via texture,” IEEE Trans. Vis. Comput. Graph. 3, 98-117 (1997).
[CrossRef]

J. Mod. Opt. (1)

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Optik (Stuttgart) (1)

M. Schulz, “Topography measurement by a reliable large-area curvature sensor,” Optik (Stuttgart) 112, 86-90 (2001).
[CrossRef]

Proc. SPIE (5)

I. Weingaertner, M. Schulz, P. Thomsen-Schmidt, and C. Elster, “Measurement of steep aspheres: a step toward nanometer accuracy,” Proc. SPIE 4449, 195-204 (2001).
[CrossRef]

P. Glenn, “Angstrom level profilometry for sub-millimeter to meter scale surface errors,” Proc. SPIE 1333, 326-336 (1990).
[CrossRef]

P. E. Glenn, “Robust, sub-angstrom level mid-spatial frequency profilometry,” Proc. SPIE 1333, 230-238 (1990).
[CrossRef]

I. Weingaertner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306-317 (1999).
[CrossRef]

W. Zou and J. Rolland, “Differential wavefront curvature sensor,” Proc. SPIE 5869, 5869171 (2005).

Publ. Astron. Soc. Pac. (1)

A. Tokovinin, “From differential image motion to seeing,” Publ. Astron. Soc. Pac. 114, 1156-1166 (2002).
[CrossRef]

Other (5)

W. Zou and J. Rolland, “Differential Shack-Hartmann curvature sensor,” U.S. patent 7,390,999 (24 June 2008).

W. Zou, “Optimization of zonal wavefront estimation and curvature measurements,” Ph.D. dissertation (University of Central Florida, 2007).

Adaptive Optics Associates Inc., part no. 1790-90-s, http://www.aoainc.com/index.html.

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

J. J. Koenderink, Solid Shape (MIT, 1990), pp. 210, 214, 212, 228, 232.

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Figures (4)

Fig. 1
Fig. 1

Concept of the Shack–Hartmann wavefront sensor.

Fig. 2
Fig. 2

The x- and y-differential shears of the Hartmann grid.

Fig. 3
Fig. 3

Experimental system of a possible implementation for the differential Shack–Hartmann curvature sensor: (a) (color online) optical layout, (b) Hartmann sampling grid, (c) (color online) picture of the experimental system.

Fig. 4
Fig. 4

Map of the principal curvatures and principal directions on a wavefront estimated from the measured slope data (arrow length scale = 0.6 ). Solid arrow lines, first principal curvatures and directions; dotted arrow lines, second principal curvatures and directions.

Equations (23)

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W x i = x i mea x i ref f ,
W y i = y i mea y i ref f ,
c x x ( i ) = 2 W x 2 i = 1 s x ( W x i W x i ) = 1 f ( x i mea x i mea s x ) c 0 , x x ( i ) ,
c y y ( i ) = 2 W y 2 i = 1 s y ( W y i W y i ) = 1 f ( y i mea y i mea s y ) c 0 , y y ( i ) ,
c 0 , x x ( i ) = 1 f ( x i Ref x i Ref s x ) ,
c 0 , y y ( i ) = 1 f ( y i Ref y i Ref s y ) .
c x y ( i ) = 2 W x y i = 1 s x ( W y i W y i ) = 1 f ( y i mea y i mea s x ) c 0 , x y ( i ) ,
c y x ( i ) = 2 W y x i = 1 s y ( W x i W x i ) = 1 f ( x i mea x i mea s y ) c 0 , y x ( i ) ,
c 0 , x y ( i ) = 1 f ( y i Ref y i Ref s x ) ,
c 0 , y x ( i ) = 1 f ( x i Ref x i Ref s y ) .
X = x e 1 + y e 2 + W ( x , y ) e 3 ,
II = ( ω ̑ 1 13 ω ̑ 1 23 ω ̑ 2 13 ω ̑ 2 23 ) ,
II = ( c x x ( i ) c y x ( i ) c x y ( i ) c y y ( i ) ) , i = 1 , 2 , , m .
II = P T II P ,
P = [ cos θ sin θ sin θ cos θ ] ,
II = ( κ 1 ( i ) 0 0 κ 2 ( i ) ) ,
κ 1 , 2 ( i ) = c x ( i ) + c y ( i ) ± ( c x ( i ) c y ( i ) ) 2 + 4 c x y ( i ) 2 2 ,
θ ( i ) = 1 2 tan 1 ( 2 c x y ( i ) c x ( i ) c y ( i ) ) .
det ( κ I II ) = 0 .
c x ( i ) = κ 1 ( i ) cos 2 [ θ ( i ) ] + κ 2 ( i ) sin 2 [ θ ( i ) ] .
cos 2 [ θ ( i ) ] = 2 c x ( i ) 2 H ( i ) κ 1 ( i ) κ 2 ( i ) ,
θ ( i ) = 1 2 cos 1 ( c x ( i ) c y ( i ) κ 1 ( i ) κ 2 ( i ) ) .
θ ( i ) = 1 2 cos 1 ( c x ( i ) c y ( i ) ( c x ( i ) c y ( i ) ) 2 + 4 c x y ( i ) 2 ) ,

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