Abstract

If the propagation of a light field can be satisfactorily described by a diffraction integral with an ABCD kernel, the propagation of its irradiance centroid is completely determined by the corresponding ABCD ray-transfer matrix in exactly the same way as if the centroid path were a conventional geometrical ray. However, potentially significant deviations from this geometrical propagation rule may arise in the presence of finite or nonuniform apertures truncating or otherwise modifying the input beam irradiance distribution.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2011 (1)

J. Ares, J. Arines, and S. Bará, “Finite-area centroid propagation in homogeneous media and range of validity of the optical Ehrenfest's theorem,” Opt. Commun. 284, 2455–2459 (2011).
[CrossRef]

2010 (1)

2007 (1)

2006 (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

2003 (1)

2002 (1)

2000 (2)

1999 (3)

1998 (4)

1997 (4)

J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

H. J. Tiziani and J. H. Chen, “Shack–Hartmann sensor for fast infrared wavefront testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).
[CrossRef]

1996 (3)

1995 (1)

1994 (2)

1992 (1)

1990 (1)

1982 (1)

1975 (1)

1927 (1)

P. Ehrenfest, “Notes on the approximate validity of quantum mechanics,” Z. Phys. 45, 455–457 (1927) (in German).
[CrossRef]

Ares, J.

J. Ares, J. Arines, and S. Bará, “Finite-area centroid propagation in homogeneous media and range of validity of the optical Ehrenfest's theorem,” Opt. Commun. 284, 2455–2459 (2011).
[CrossRef]

J. Arines and J. Ares, “Minimum variance centroid thresholding,” Opt. Lett. 27, 497–499 (2002).
[CrossRef]

J. Ares, T. Mancebo, and S. Bará, “Position and displacement sensing with Shack–Hartmann wavefront sensors,” Appl. Opt. 39, 1511–1520 (2000).
[CrossRef]

Arines, J.

J. Ares, J. Arines, and S. Bará, “Finite-area centroid propagation in homogeneous media and range of validity of the optical Ehrenfest's theorem,” Opt. Commun. 284, 2455–2459 (2011).
[CrossRef]

J. Arines and J. Ares, “Minimum variance centroid thresholding,” Opt. Lett. 27, 497–499 (2002).
[CrossRef]

Artal, P.

Artzner, G.

G. Artzner, “Aspherical wavefront measurements: Shack–Hartmann numerical and practical experiments,” Pure Appl. Opt. 7, 435–448 (1998).
[CrossRef]

Bagini, V.

Bará, S.

Bille, J.

Blumel, T.

Boreman, G. D.

Born, M.

M. Born and E. Wolf, “Appendix IV,” in Principles of Optics(Pergamon, 1987), pp. 755–759.

Brooks, A. F.

Burow, R.

Chen, J. H.

H. J. Tiziani and J. H. Chen, “Shack–Hartmann sensor for fast infrared wavefront testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

Cook, R. J.

Dainty, C.

Dainty, J. C.

Dayton, D.

Dorronsoro, C.

Doyle, S. M.

N. S. Prasad, S. M. Doyle, and M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Ehrenfest, P.

P. Ehrenfest, “Notes on the approximate validity of quantum mechanics,” Z. Phys. 45, 455–457 (1927) (in German).
[CrossRef]

Elssner, K.-E.

Fontanella, J. C.

Fusco, T.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

Giles, M. K.

N. S. Prasad, S. M. Doyle, and M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Goelz, S.

Gonglewski, J.

Grimm, B.

Irwan, R.

Jiang, W.

Kelly, T. L.

Kohno, T.

T. Kohno and S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

Lane, R. G.

Leroux, C.

Liang, J.

Lindlein, N.

Ling, N.

Loos, G.

Losada, M. A.

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

Mancebo, T.

Merkle, F.

F. Merkle, “'Adaptive optics,” in International Trends in Optics (Academic, 1991), pp. 375–390.

Michau, V.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

Montera, D. A.

Moreno, E.

Moreno-Barriuso, E.

Munch, J.

Navarro, R.

Nicholls, T. W.

Nicolle, M.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

Palma, C.

Pfund, J.

Pierson, B.

Prasad, N. S.

N. S. Prasad, S. M. Doyle, and M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Prieto, P. M.

Primot, J.

Rao, C.

Roggemann, M. C.

Rousset, G.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

Ruck, D. W.

Ruggiu, J.-M.

Schwider, J.

Siegman, A. E.

A. E. Siegman, “Complex paraxial wave optics,” in Lasers(University Science Books, 1986), Chap. 20, pp. 777–782.

Silbaugh, E. E.

Solomon, C. J.

Spielbusch, B.

Tanaka, S.

T. Kohno and S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

Tatarskii, V. I.

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, 1967), pp. 385–390 (in Russian).

Teague, M. R.

Thomas, S.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

Tiziani, H. J.

H. J. Tiziani and J. H. Chen, “Shack–Hartmann sensor for fast infrared wavefront testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

Tokovinin, A.

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

Tyson, R. K.

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

Vargas-Martin, F.

Veitch, P. J.

Welsh, B. M.

Williams, D. R.

Wolf, E.

M. Born and E. Wolf, “Appendix IV,” in Principles of Optics(Pergamon, 1987), pp. 755–759.

Appl. Opt. (4)

J. Mod. Opt. (1)

H. J. Tiziani and J. H. Chen, “Shack–Hartmann sensor for fast infrared wavefront testing,” J. Mod. Opt. 44, 535–541 (1997).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

S. Bará, “Measuring eye aberrations with Hartmann–Shack wavefront sensors: Should the irradiance distribution across the eye pupil be taken into account?” J. Opt. Soc. Am. A 20, 2237–2245 (2003).
[CrossRef]

C. Palma and V. Bagini, “Extension of the Fresnel transform to ABCD systems,” J. Opt. Soc. Am. A 14, 1774–1779 (1997).
[CrossRef]

E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wavefront slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460(1996).
[CrossRef]

J. Primot, G. Rousset, and J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

J. Liang, B. Grimm, S. Goelz, and J. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wavefront sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

P. M. Prieto, F. Vargas-Martin, S. Goelz, and P. Artal, “Analysis of the performance of the Hartmann–Shack sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000).
[CrossRef]

R. Navarro, E. Moreno, and C. Dorronsoro, “Monochromatic aberrations and point-spread functions of the human eye across the visual field,” J. Opt. Soc. Am. A 15, 2522–2529 (1998).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

S. Thomas, T. Fusco, A. Tokovinin, M. Nicolle, V. Michau, and G. Rousset, “Comparison of centroid computation algorithms in a Shack–Hartmann sensor,” Mon. Not. R. Astron. Soc. 371, 323–336 (2006)
[CrossRef]

Opt. Commun. (1)

J. Ares, J. Arines, and S. Bará, “Finite-area centroid propagation in homogeneous media and range of validity of the optical Ehrenfest's theorem,” Opt. Commun. 284, 2455–2459 (2011).
[CrossRef]

Opt. Eng. (1)

N. S. Prasad, S. M. Doyle, and M. K. Giles, “Collimation and beam alignment: testing and estimation using liquid-crystal televisions,” Opt. Eng. 35, 1815–1819 (1996).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Opt. Rev. (1)

T. Kohno and S. Tanaka, “Figure measurement of concave mirror by fiber-grating Hartmann test,” Opt. Rev. 1, 118–120 (1994).
[CrossRef]

Optom. Vision Sci. (1)

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vision Sci. 74, 540–547 (1997).
[CrossRef]

Pure Appl. Opt. (1)

G. Artzner, “Aspherical wavefront measurements: Shack–Hartmann numerical and practical experiments,” Pure Appl. Opt. 7, 435–448 (1998).
[CrossRef]

Z. Phys. (1)

P. Ehrenfest, “Notes on the approximate validity of quantum mechanics,” Z. Phys. 45, 455–457 (1927) (in German).
[CrossRef]

Other (5)

A. E. Siegman, “Complex paraxial wave optics,” in Lasers(University Science Books, 1986), Chap. 20, pp. 777–782.

M. Born and E. Wolf, “Appendix IV,” in Principles of Optics(Pergamon, 1987), pp. 755–759.

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

F. Merkle, “'Adaptive optics,” in International Trends in Optics (Academic, 1991), pp. 375–390.

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, 1967), pp. 385–390 (in Russian).

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Figures (4)

Fig. 1
Fig. 1

Output irradiance distribution for an input Gaussian beam of half-width σ = R (measured at the 1 / e irradiance level), centered at x = 0 , y = R , and truncated by a lens with circular pupil of radius R.

Fig. 2
Fig. 2

Output versus input centroid position for Gaussian beams with the half-width σ values indicated in curve labels. The unlabeled diagonal straight line corresponds to the ideal ABCD behavior. All units normalized to the lens pupil radius, R.

Fig. 3
Fig. 3

Output momentum (mrad) versus input centroid position for the wavefronts in Fig. 2 after being refracted by an f / 5 unaberrated converging lens. The unlabeled straight line corresponds to the ideal ABCD behavior. Length units normalized to the lens radius R.

Fig. 4
Fig. 4

Longitudinal section of the image space showing the centroid propagation paths after beam refraction by a f / 5 converging lens assuming the incident wavefront has its input centroid located at the lens pupil rim. Horizontal line, optical axis; slanted dotted line, ideal ABCD behavior; solid lines, actual centroid paths. Labels correspond to the beam half-width values σ. All centroid paths intersect at the image of the center of curvature of the incoming wavefont (in this example the back focal point of the lens at z = 10 ). All distances normalized to R.

Equations (20)

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

u o ( r o ) = K ( r o , r ) u ( r ) d 2 r ,
K ( r o , r ) = { ( 1 i λ B ) exp [ i π λ B ( A r 2 + D r o 2 2 r · r o ) ] ( B 0 ) 1 A exp ( i π C λ A r o 2 ) δ ( r r o A ) ( B = 0 ) ,
x o = x ^ o = E o 1 u o * ( r o ) r o u o ( r o ) d 2 r o = E o 1 I o ( r o ) r o d 2 r o ,
p o = p ^ o = E o 1 u o * ( r o ) ( i k ) 1 o u o ( r o ) d 2 r o = E o 1 I o ( r o ) o W o ( r o ) d 2 r o ,
( x o p o ) = ( A B C D ) ( x p ) ,
( x o p o ) = ( x p ) + ( E 1 [ ( E / E o ) T ( r ) 1 ] I ( r ) r d 2 r E 1 [ ( E / E o ) T ( r ) 1 ] I ( r ) W ( r ) d 2 r + E o 1 T ( r ) I ( r ) W t ( r ) d 2 r ) .
( Δ x Δ p ) = ( E 1 [ ( E / E o ) T ( r ) 1 ] I ( r ) r d 2 r E 1 [ ( E / E o ) T ( r ) 1 ] I ( r ) W ( r ) d 2 r )
( x o p o ) = ( 1 0 1 / f 1 ) ( x + Δ x p + Δ p ) ,
( x o p o ) = ( 1 0 1 / f 1 ) ( x p ) + ( Δ x o Δ p o ) ,
( Δ x o Δ p o ) = ( 1 0 1 / f 1 ) ( Δ x Δ p ) .
( x o ( z ) p o ( z ) ) = ( 1 z 0 1 ) ( x o p o ) .
( Δ x o ( z ) Δ p o ( z ) ) = ( 1 z 0 1 ) ( Δ x o Δ p o ) ,
Δ x o ( z ) = [ 1 + z ( 1 / f + 1 / s ) ] Δ x Δ p o ( z ) = ( 1 / f + 1 / s ) Δ x .
δ ( x ) = 1 2 π exp [ i α x ] d α ,
δ ( n ) ( x ) d n δ ( x ) d x n = 1 2 π ( i ) n α n exp [ i α x ] d α ,
α n exp [ i α x ] d α = 2 π i n δ ( n ) ( x ) .
f ( x ) δ ( n ) ( x ) d x = ( 1 ) n f ( n ) ( 0 ) ,
f ( x ) δ ( n ) [ c ( x x ) ] d x = ( 1 ) n | c | 1 f ( n ) ( x ) .
u o ( r o ) = 1 A exp ( i π C λ A r o 2 ) u ( r o A ) ,
a o ( r o ) = 1 A a ( r o A ) W o ( r o ) = C 2 A r o 2 + W ( r o A ) ,

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