Abstract

A novel differential Hartmann sensor is described. It can be used to determine the characteristics of an optic accurately, precisely, and simply without detailed knowledge of the wavefront used to illuminate the optical system or of the geometry of the measurement system. We demonstrate the application of this sensor to both zonal and modal optical testing of lenses. We also describe a dual-camera implementation of the sensor that would enable high-speed optical testing.

© 2007 Optical Society of America

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References

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  1. I. Ghozeil, "Hartmann and other screen tests," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), Chap. 10, pp. 367-396.
  2. F. Roddier, "Variations on a Hartmann theme," Opt. Eng. 29, 1239-1242 (1990).
    [CrossRef]
  3. D. Malacara and Z. Malacara, "Testing and centering of lenses by means of a Hartmann test with four holes," Opt. Eng. 31, 1551-1555 (1992).
    [CrossRef]
  4. C. Castellini, F. Francini, and B. Tiribilli, "Hartmann test modification for measuring ophthalmic progressive lenses," Appl. Opt. 33, 4120-4124 (1994).
    [CrossRef] [PubMed]
  5. V. Laude, S. Olivier, C. Dirson, and J.-P. Huignard, "Hartmann wave-front scanner," Opt. Lett. 24, 1796-1798 (1999).
    [CrossRef]
  6. D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
    [CrossRef]
  7. M. Servin, F. J. Cuevas, D. Malacara, and J. L. Marroquin, "Direct ray aberration estimation in Hartmanngrams by use of a regularized phase-tracking system," Appl. Opt. 38, 2862-2869 (1999).
    [CrossRef]
  8. R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).
  9. M. Abitol, A. Blum, A. Halimi, and E. Meimoun, "Apparatus for mapping optical elements," U.S. patent 5,825,476 (20 October 1998).
  10. J. Pfund, N. Lindlein, and J. Schwider, "Dynamic range expansion of a Shack-Hartmann sensor by use of a modified unwrapping algorithm," Opt. Lett. 23, 995-997 (1998).
    [CrossRef]
  11. M. A. van Dam, D. L. Mignant, and B. A. Macintosh, "Performance of the Keck Observatory adaptive optics system," Appl. Opt. 43, 5458-5467 (2004).
    [CrossRef] [PubMed]
  12. A. F. Brooks, P. Veitch, and J. Munch, "Ultra-sensitive wavefront measurement using a Hartmann sensor," presented at the Australian Institute of Physics 17th National Congress, Brisbane, Australia, 3-7 December 2006.
  13. W. H. Southwell, "Wave-front estimation from wave-front slope measurements," J. Opt. Soc. Am. 70, 998-1006 (1980).
    [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), pp. 59-70.
  15. T.-L. Kelly, A. F. Brooks, P. J. Veitch, and J. Munch, "Centroid estimation for Hartmann wavefront sensors," manuscript in preparation.

2006

D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
[CrossRef]

2004

1999

1998

1994

1992

D. Malacara and Z. Malacara, "Testing and centering of lenses by means of a Hartmann test with four holes," Opt. Eng. 31, 1551-1555 (1992).
[CrossRef]

1990

F. Roddier, "Variations on a Hartmann theme," Opt. Eng. 29, 1239-1242 (1990).
[CrossRef]

1980

Abitol, M.

M. Abitol, A. Blum, A. Halimi, and E. Meimoun, "Apparatus for mapping optical elements," U.S. patent 5,825,476 (20 October 1998).

Almazán-Cuellar, S.

D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
[CrossRef]

Blum, A.

M. Abitol, A. Blum, A. Halimi, and E. Meimoun, "Apparatus for mapping optical elements," U.S. patent 5,825,476 (20 October 1998).

Brooks, A. F.

A. F. Brooks, P. Veitch, and J. Munch, "Ultra-sensitive wavefront measurement using a Hartmann sensor," presented at the Australian Institute of Physics 17th National Congress, Brisbane, Australia, 3-7 December 2006.

T.-L. Kelly, A. F. Brooks, P. J. Veitch, and J. Munch, "Centroid estimation for Hartmann wavefront sensors," manuscript in preparation.

Castellini, C.

Cuevas, F. J.

Dirson, C.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), pp. 59-70.

Francini, F.

Ghozeil, I.

I. Ghozeil, "Hartmann and other screen tests," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), Chap. 10, pp. 367-396.

González-Silva, J. A.

D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
[CrossRef]

Halimi, A.

M. Abitol, A. Blum, A. Halimi, and E. Meimoun, "Apparatus for mapping optical elements," U.S. patent 5,825,476 (20 October 1998).

Huignard, J.-P.

Kelly, T.-L.

T.-L. Kelly, A. F. Brooks, P. J. Veitch, and J. Munch, "Centroid estimation for Hartmann wavefront sensors," manuscript in preparation.

Laude, V.

Lindlein, N.

Macintosh, B. A.

Malacara, D.

Malacara, Z.

D. Malacara and Z. Malacara, "Testing and centering of lenses by means of a Hartmann test with four holes," Opt. Eng. 31, 1551-1555 (1992).
[CrossRef]

Marroquin, J. L.

Meimoun, E.

M. Abitol, A. Blum, A. Halimi, and E. Meimoun, "Apparatus for mapping optical elements," U.S. patent 5,825,476 (20 October 1998).

Mignant, D. L.

Munch, J.

T.-L. Kelly, A. F. Brooks, P. J. Veitch, and J. Munch, "Centroid estimation for Hartmann wavefront sensors," manuscript in preparation.

A. F. Brooks, P. Veitch, and J. Munch, "Ultra-sensitive wavefront measurement using a Hartmann sensor," presented at the Australian Institute of Physics 17th National Congress, Brisbane, Australia, 3-7 December 2006.

Olivier, S.

Pfund, J.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), pp. 59-70.

Roddier, F.

F. Roddier, "Variations on a Hartmann theme," Opt. Eng. 29, 1239-1242 (1990).
[CrossRef]

Salas-Peimbert, D. P.

D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
[CrossRef]

Schwider, J.

Servin, M.

Southwell, W. H.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), pp. 59-70.

Tiribilli, B.

Trujillo-Schiaffino, G.

D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

van Dam, M. A.

Veitch, P.

A. F. Brooks, P. Veitch, and J. Munch, "Ultra-sensitive wavefront measurement using a Hartmann sensor," presented at the Australian Institute of Physics 17th National Congress, Brisbane, Australia, 3-7 December 2006.

Veitch, P. J.

T.-L. Kelly, A. F. Brooks, P. J. Veitch, and J. Munch, "Centroid estimation for Hartmann wavefront sensors," manuscript in preparation.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), pp. 59-70.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Eng.

F. Roddier, "Variations on a Hartmann theme," Opt. Eng. 29, 1239-1242 (1990).
[CrossRef]

D. Malacara and Z. Malacara, "Testing and centering of lenses by means of a Hartmann test with four holes," Opt. Eng. 31, 1551-1555 (1992).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

D. P. Salas-Peimbert, G. Trujillo-Schiaffino, J. A. González-Silva, and S. Almazán-Cuellar, "Simple Hartmann test data interpretation for ophthalmic lenses," Rev. Sci. Instrum. 77, 043102 (2006).
[CrossRef]

Other

A. F. Brooks, P. Veitch, and J. Munch, "Ultra-sensitive wavefront measurement using a Hartmann sensor," presented at the Australian Institute of Physics 17th National Congress, Brisbane, Australia, 3-7 December 2006.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), pp. 59-70.

T.-L. Kelly, A. F. Brooks, P. J. Veitch, and J. Munch, "Centroid estimation for Hartmann wavefront sensors," manuscript in preparation.

R. K. Tyson, Principles of Adaptive Optics (Academic, 1991).

M. Abitol, A. Blum, A. Halimi, and E. Meimoun, "Apparatus for mapping optical elements," U.S. patent 5,825,476 (20 October 1998).

I. Ghozeil, "Hartmann and other screen tests," in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), Chap. 10, pp. 367-396.

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Figures (5)

Fig. 1
Fig. 1

Schematic layout of the differential Hartmann testing system. Only one of the rays produced by the Hartmann plate is traced through the optical system. When the unknown optic is placed in the path of the Hartmann ray, the ray is deflected to produce the “aberrated ray.” A possible deflection is shown. The CCD camera is positioned at the locations denoted CCD 1 and CCD 2, to yield two transverse aberrations.

Fig. 2
Fig. 2

Schematic of a Hartmann plate with a 6 × 6 square array of holes. Dividing the holes into 3 × 3 subsets, one of which is shown, and using a spot-centered analysis would allow the optical parameters to be determined at 16 locations. Including non-spot-centered subsets would allow the parameters to be calculated at other locations.

Fig. 3
Fig. 3

(a) Plot of the 17 locations at which the zonal spherical power of the microscope objective lens was calculated. The central location is near the optical axis of the lens. (b) A contour map of the spherical power of a microscope objective lens. The zonal measurements of spherical power were transformed into a 10 × 10 regular grid with smoothness parameter = 0.6 using the correlation (Kriging) method in the Microcal Origin package. The 0.1 diopter contour interval is 3 times larger than the reproducibility of the measurements used to generate the plot.

Fig. 4
Fig. 4

Schematic layout of the differential Hartmann sensor used for the modal analysis of the reference lenses. A beam splitter and two CMOS sensors were used to implement the differential measurement. A fiber-coupled (F-C) SLED light source was used to prevent unwanted interference fringes. The demagnifying lens, L2, reduces the size of the Hartmann spot pattern to that of the CMOS sensors.

Fig. 5
Fig. 5

Images of Hartmann spot patterns for a 4 diopter lens with δ l sensors = 9.20 mm , R 1 = 0.360 , and R 2 = 0.450 : (a), (b) reference spots at planes 1 and 2; (c), (d) “aberrated” spot at planes 1 and 2, located after the focal plane of the 4 diopter lens.

Tables (1)

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Table 1 Measured and Nominal Values of S , C , a and P

Equations (33)

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T A x = l W x = i = 0 m 1 a i X i ,     T A y = l W y = i = 0 m 1 a i Y i ,
X i = Ψ i x ,     Y i = Ψ i y ,     a i = α i l ,
χ 2 = j = 1 N { [ T A x j i = 0 m 1 a i X i ( x j , y j ) ] 2 + [ T A y j i = 0 m 1 a i Y i ( x j , y j ) ] 2 } ,
M a = b ,
M = X T X + Y T Y ,
X = [ X 0 ( x 1 , y 1 ) X m 1 ( x 1 , y 1 ) X 0 ( x N , y N ) X m 1 ( x N , y N ) ] ,
Y = [ Y 0 ( x 1 , y 1 ) Y m 1 ( x 1 , y 1 ) Y 0 ( x N , y N ) Y m 1 ( x N , y N ) ] ,
a = [ a 0 a m 1 ] ,
b = X T b x + Y T b y ,
b x = [ T A x 1 T A x N ] ,     b y = [ T A y 1 T A y N ] .
α i = ( a i 1 a i 2 ) / δ l ,
W = P ( x cos α + y sin α ) + 0.5 S ( x 2 + y 2 ) + 0.5 C ( x sin ϕ y cos ϕ ) 2 + B ( x cos β + y sin β ) × ( x 2 + y 2 ) + A ( x 2 + y 2 ) 2 ,
T A x = a 0 + a 1 x + a 2 y + 2 a 3 x y + 3 a 4 x 2 + a 4 y 2 + a 5 x 3 + a 5 x y 2 ,
T A y = a 6 + a 2 x + a 7 y + 2 a 4 x y + a 3 x 2 + 3 a 3 y 2 + a 5 y 3 + a 5 x 2 y ,
a 0 = l P cos α , a 1 = l ( S + C sin 2 ϕ ) ,
a 2 = l ( C sin ϕ cos ϕ ) , a 3 = l B sin β ,
a 4 = l B cos β , a 5 = l 4 A ,
a 6 = l P sin α , a 7 = l ( S + C cos 2 ϕ ) .
χ 2 = j = 1 N { [ T A x j ( a 0 + a 1 x j + a 2 y j + 2 a 3 x j y j + 3 a 4 x j 2 + a 4 y j 2 + a 5 x j 3 + a 5 x j y j 2 ) ] 2 + [ T A y j ( a 6 + a 2 x j + a 7 y j + 2 a 4 x j y j + a 3 x j 2 + 3 a 3 y j 2 + a 5 y j 3 + a 5 x j 2 y j ) ] 2 } ,
X 0 = 1 , Y = 0 , X 1 = x , Y 1 = 0 , X 2 = y , Y 2 = x , X 3 = 2 x y , Y 3 = x 2 + 3 y 2 , X 4 = 3 x 2 + y 2 , Y 4 = 2 x y , X 5 = x 3 + x y 2 , Y 5 = y 3 + x 2 y , X 6 = 0 , Y 6 = 1 , X 7 = 0 , Y 7 = y .
tan α = a 6 / a 0 ,
tan 2 ϕ = 2 a 2 / ( a 1 a 7 ) ,
tan β = a 3 / a 4 ,
P l = ( a 0 2 + a 6 2 ) 1 / 2 ,
C l = { ( a 1 a 7 ) / cos 2 ϕ , cos 2 ϕ 0 a 2 / cos ϕ sin ϕ , cos 2 ϕ = 0 ,
S l = ( a 1 + a 7 + C l ) / 2 ,
B l = ( a 3 2 + a 4 2 ) 1 / 2 ,
A l = a 5 / 4 .
P = ( P l 2 P l 1 ) / δ l ,
C = ( C l 2 C l 1 ) / δ l ,
S = ( S l 2 S l 1 ) / δ l ,
B = ( B l 2 B l 1 ) / δ l ,
A = ( A l 2 A l 1 ) / δ l ,

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