Abstract

In Ref. [1], it was demonstrated that the significant systematic errors of a type of large dynamic range aberrometer are strongly related to the power error (defocus) in the input wavefront. In this paper, a generalized theoretical analysis based on vector aberration theory is presented, and local shift errors of the SH spot pattern as a function of the lenslet position and the local wavefront tilt over the corresponding lenslet are derived. Three special cases, a spherical wavefront, a crossed cylindrical wavefront, and a cylindrical wavefront, are analyzed and the possibly affected Zernike terms in the wavefront reconstruction are investigated. The simulation and experimental results are illustrated to verify the theoretical predictions.

© 2009 Optical Society of America

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References

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  1. P. Wu, E. DeHoog, and J. Schwiegerling, “Systematic error of a large dynamic range aberrometer,” Appl. Opt. (to be published).
  2. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann-Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949-1957 (1994).
    [CrossRef]
  3. J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE Press, 2004).
    [CrossRef]
  4. J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).
    [CrossRef]
  5. R. Shack, class notes of Opti518 at Optical Science College, University of Arizona.
  6. W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986).
  7. J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33 (6), 2045-2061(1994).
    [CrossRef]
  8. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005).
    [CrossRef]
  9. W. Kaplan, “The directional derivative,” in Advanced Calculus, 4th ed. (Addison-Wesley, 1991), pp 135-138.
  10. L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eye,” in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC1.

2005

1994

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eye,” in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC1.

Bille, J. F.

DeHoog, E.

P. Wu, E. DeHoog, and J. Schwiegerling, “Systematic error of a large dynamic range aberrometer,” Appl. Opt. (to be published).

Geary, J. M.

J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).
[CrossRef]

Goelz, S.

Grimm, B.

Kaplan, W.

W. Kaplan, “The directional derivative,” in Advanced Calculus, 4th ed. (Addison-Wesley, 1991), pp 135-138.

Liang, J.

Sasian, J.

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33 (6), 2045-2061(1994).
[CrossRef]

Schwiegerling, J.

P. Wu, E. DeHoog, and J. Schwiegerling, “Systematic error of a large dynamic range aberrometer,” Appl. Opt. (to be published).

J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE Press, 2004).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eye,” in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC1.

Shack, R.

R. Shack, class notes of Opti518 at Optical Science College, University of Arizona.

Thibos, L. N.

L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eye,” in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC1.

Thompson, K. P.

Webb, R.

L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eye,” in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC1.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986).

Wu, P.

P. Wu, E. DeHoog, and J. Schwiegerling, “Systematic error of a large dynamic range aberrometer,” Appl. Opt. (to be published).

J. Opt. Soc. Am. A

Opt. Eng.

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33 (6), 2045-2061(1994).
[CrossRef]

Other

P. Wu, E. DeHoog, and J. Schwiegerling, “Systematic error of a large dynamic range aberrometer,” Appl. Opt. (to be published).

W. Kaplan, “The directional derivative,” in Advanced Calculus, 4th ed. (Addison-Wesley, 1991), pp 135-138.

L. N. Thibos, R. A. Applegate, J. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eye,” in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, 2000), paper SuC1.

J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE Press, 2004).
[CrossRef]

J. M. Geary, Introduction to Wavefront Sensors (SPIE Press, 1995).
[CrossRef]

R. Shack, class notes of Opti518 at Optical Science College, University of Arizona.

W. T. Welford, Aberrations of Optical Systems (Taylor & Francis, 1986).

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Figures (6)

Fig. 1
Fig. 1

Ideal components and their roles in the whole system are illustrated. The first part is the afocal pupil relay optics, by which the eye pupil is conjugated to the lenslet array. The second part is the lenslet array, by which the wavefront is sampled and the SH spots are focused at the intermediate image plane. The third part is the imaging relay optics; the intermediate image of SH spots is supposed to be ideally imaged with a specified magnification.

Fig. 2
Fig. 2

Chief ray associated with the specified field of angle and lenslet, which are described by the normalized field vector H and the normalized aperture vector r , respectively, is demonstrated. y is the vector describing the ideal intercept at the back focal plane of the chief ray and ε is the transverse ray aberration associated with y . Likewise, y is the vector describing the ideal intercept at the image plane of the chief ray and ε is the transverse ray aberration associated with y . f lenslet is the focal length of the lenslet, z is the distance from the front principal plane of the imaging lens to the intermediate SH spot image plane, z is the distance from the rear principal plane to the final image plane, and f is the rear focal length of the imaging lens.

Fig. 3
Fig. 3

Directional derivatives of the wavefront at point ( r , θ ) are illustrated. (a) The directional derivatives of a spherical wavefront in the ρ direction and its orthogonal direction j ; the latter component in this case is consistently zero. (b) The directional derivatives of a toroidal wavefront in directions denoted as x and y , respectively; both of them are then projected onto the ρ direction and the j direction.

Fig. 4
Fig. 4

In case the incident wavefront with various sphere powers is measured, the measured Z 4 , 0 coefficient is a quadratic function of power factor K D .

Fig. 5
Fig. 5

In case the incident wavefront with various crossed cylindrical powers is measured, as shown in (a) and (c), the measured Z 4 , 0 and Z 4 , 4 coefficients are a function of K C 2 . (b) The measured Z 4 , 2 coefficient data fit a linear curve quite well.

Fig. 6
Fig. 6

In case the incident wavefront with various cylindrical powers is measured, as shown in (a) and (b), the measured Z 4 , 0 and Z 4 , 2 coefficients are quadratic functions of K C and (c) the measured Z 4 , 4 coefficient is a function of K C 2 .

Tables (1)

Tables Icon

Table 1 In the Three Special Cases, a Spherical Wavefront, a Crossed Cylindrical Wavefront, and a Cylindrical Wavefront, the Coefficients of Local Shift Error Components and Their Corresponding Zernike Terms in Wavefront Reconstruction are Some Functions of the Power Factor ( K D or K C )

Equations (30)

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r = x r max x ^ + y r max y ^ ,
H = H cos ϕ · ρ ^ + H sin ϕ · j ^ = H ρ ^ · ρ ^ + H j ^ · j ^ ,
W = ρ ^ W · ρ ^ + j ^ W · j ^ .
H = 1 tan θ max ( ρ ^ W · ρ ^ + j ^ W · j ^ ) ,
Δ W = W 040 ( r · r ) 2 + W 131 ( H · r ) ( r · r ) + W 220 ( H · H ) ( r · r ) + W 222 ( H · r ) 2 + W 311 ( H · r ) ( H · H ) ,
ε = R exit n r exit [ 4 W 040 ( r · r ) r + W 131 ( ( r · r ) H + 2 ( H · r ) r ) + 2 W 220 ( H · H ) r + 2 W 222 ( H · r ) r + W 311 ( H · H ) H ] ,
ε = A ρ ^ + B j ^ ,
A = R exit n r exit ( 4 W 040 r 3 + 3 W 131 H r 2 cos ϕ + 2 W 220 H 2 r cos 2 ϕ + W 311 H 3 cos ϕ ) , B = R exit n r exit ( W 131 H r 2 sin ϕ + W 222 H 2 r sin 2 ϕ + W 311 H 3 sin ϕ ) ,
ε = z f ε = ( 1 m ) ε .
ε ρ ^ ( ρ ^ W , j ^ W ; r , θ ) = C 00 r 3 + C 10 r 2 ( ρ ^ W ) + C 20 r ( ρ ^ W ) 2 + C 02 r ( j ^ W ) 2 + C 12 ( ρ ^ W ) ( j ^ W ) 2 + C 30 ( ρ ^ W ) 3 , ε j ^ ( ρ ^ W , j ^ W ; r , θ ) = C 01 r 2 ( j ^ W ) + C 11 ( ρ ^ W ) ( j ^ W ) + C 21 ( ρ ^ W ) 2 ( j ^ W ) + C 03 ( j ^ W ) 3 ,
C 00 = 4 C · W 040 , C 10 = 3 C 01 = 3 C · 1 tan θ max W 131 , C 20 = C 02 + C 11 = 2 C · ( 1 tan θ max ) 2 ( W 220 + W 222 ) , C 02 = 2 C · ( 1 tan θ max ) 2 W 220 , C 11 = 2 C · ( 1 tan θ max ) 2 W 222 , C 12 = C 30 = C 21 = C 03 = C · ( 1 tan θ max ) 3 W 311 ,
C = R exit n r exit ( 1 m ) .
ρ ^ W ( r , θ ) = K D r , j ^ W ( r , θ ) = 0.
ε ρ ^ ( ρ ^ W , j ^ W ; r , θ ) = ( C 00 + C 10 K D + C 20 K D 2 + C 30 K D 3 ) r 3 , ε j ^ ( ρ ^ W , j ^ W ; r , θ ) = 0.
y ^ W ( r , θ ) = r sin ( θ ϕ ) R y , x ^ W ( r , θ ) = r cos ( θ ϕ ) R x .
ρ ^ W ( r , θ ) = r cos 2 ( θ ϕ ) R x + r sin 2 ( θ ϕ ) R y , j ^ W ( r , θ ) = r cos ( θ ϕ ) sin ( θ ϕ ) R x r cos ( θ ϕ ) sin ( θ ϕ ) R y .
ρ ^ W ( r , θ ) = K C r cos 2 ( θ ϕ ) , j ^ W ( r , θ ) = K C r sin 2 ( θ ϕ ) .
ε ρ ^ ( ρ ^ W , j ^ W ; r , θ ) = A 0 r 3 + A 1 r 3 cos 2 ( θ ϕ ) + A 2 r 3 cos 4 ( θ ϕ ) = A 0 r 3 + A 1 cos 2 ϕ · r 3 cos 2 θ + A 1 sin 2 ϕ · r 3 sin 2 θ + A 2 cos 4 ϕ · r 3 cos 4 θ + A 1 sin 4 ϕ · r 3 sin 4 θ , ε j ^ ( ρ ^ W , j ^ W ; r , θ ) = B 1 r 3 sin 2 ( θ ϕ ) + B 2 r 3 sin 4 ( θ ϕ ) = B 1 cos 2 ϕ · r 3 sin 2 θ B 1 sin 2 ϕ · r 3 cos 2 θ + B 2 cos 4 ϕ · r 3 sin 4 θ B 2 sin 4 ϕ · r 3 cos 4 θ ,
A 0 = C 00 + 1 2 ( C 20 + C 02 ) K C 2 , A 1 = C 10 K C + C 30 K C 3 , A 2 = 1 2 C 20 ( K C 2 ) + ( 1 2 ) C 02 ( K C 2 ) = 1 2 C 11 ( K C 2 ) , B 1 = C 01 K C + C 30 K C 3 , B 2 = 1 2 C 11 ( K C 2 ) .
ε ρ ^ ( ρ ^ W , j ^ W ; r , θ ) = A 0 r 3 + A 1 r 3 cos 2 θ + A 2 r 3 cos 4 θ , ε j ^ ( ρ ^ W , j ^ W ; r , θ ) = B 1 r 3 sin 2 θ + B 2 r 3 sin 4 θ .
ρ ^ W ( r , θ ) = r cos 2 ( θ ϕ ) R x = K C r ( cos 2 ( θ ϕ ) + 1 2 ) , j ^ W ( r , θ ) = r cos ( θ ϕ ) sin ( θ ϕ ) R x = K C r ( sin 2 ( θ ϕ ) 2 ) .
ε ρ ^ ( ρ ^ W , j ^ W ; r , θ ) = A 0 r 3 + A 1 r 3 cos 2 ( θ ϕ ) + A 2 r 3 cos 4 ( θ ϕ ) , ε j ^ ( ρ ^ W , j ^ W ; r , θ ) = B 1 r 3 sin 2 ( θ ϕ ) + B 2 r 3 sin 4 ( θ ϕ ) ,
A 0 = C 00 + 1 2 C 10 K C + 3 8 C 20 ( K C 2 ) + 1 8 C 02 ( K C 2 ) + 3 8 C 30 ( K C 3 ) , A 1 = 1 2 C 10 K C + 1 2 C 20 ( K C 2 ) + 1 2 C 30 ( K C 3 ) , A 2 = 1 8 C 20 ( K C 2 ) + ( 1 8 ) C 02 ( K C 2 ) + 1 8 C 30 ( K C 3 ) = C 11 ( 1 8 K C 2 ) + C 30 ( 1 8 K C 3 ) , B 1 = 1 2 C 01 ( K C ) + 1 4 C 11 ( K C 2 ) + 1 8 C 30 ( K C 3 ) , B 2 = 1 8 C 11 ( K C 2 ) + 1 8 C 30 ( K C 3 ) .
ε ρ ^ ( ρ ^ W , j ^ W ; r , θ ) = A 0 r 3 + A 1 r 3 cos 2 θ + A 2 r 3 cos 4 θ , ε j ^ ( ρ ^ W , j ^ W ; r , θ ) = B 1 r 3 sin 2 θ + B 2 r 3 sin 4 θ .
[ ρ ^ Z j ^ Z ] = [ cos θ sin θ sin θ cos θ ] [ Z x Z y ] .
[ ρ ^ Z 4 , 4 j ^ Z 4 , 4 ] = [ ρ ^ 10 r 4 cos 4 θ j ^ 10 r 4 cos 4 θ ] = [ 4 10 r 3 cos 4 θ 4 10 r 3 sin 4 θ ] ,
[ ρ ^ Z 4 , 2 j ^ Z 4 , 2 ] = [ ρ ^ 10 ( 4 r 4 3 r 2 ) cos 2 θ j ^ 10 ( 4 r 4 3 r 2 ) cos 2 θ ] = [ 10 ( 16 r 3 6 r ) cos 2 θ 10 ( 8 r 3 6 r ) sin 2 θ ] ,
[ ρ ^ Z 4 , 0 j ^ Z 4 , 0 ] = [ ρ ^ 5 ( 6 r 4 6 r 2 + 1 ) j ^ 5 ( 6 r 4 6 r 2 + 1 ) ] = [ 5 ( 24 r 3 12 r ) 0 ] ,
[ ρ ^ Z 4 , 2 j ^ Z 4 , 2 ] = [ ρ ^ 10 ( 4 r 4 3 r 2 ) sin 2 θ j ^ 10 ( 4 r 4 3 r 2 ) sin 2 θ ] = [ 10 ( 16 r 3 6 r ) sin 2 θ 10 ( 8 r 3 6 r ) cos 2 θ ] ,
[ ρ ^ Z 4 , 4 j ^ Z 4 , 4 ] = [ ρ ^ 10 r 4 sin 4 θ j ^ 10 r 4 sin 4 θ ] = [ 4 10 r 3 sin 4 θ 4 10 r 3 cos 4 θ ] .

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