Abstract

The Shack–Hartmann sensor uses a microlens array and a CCD camera for wave-front measurements. To obtain wave-front measurements with high accuracy, an accurate relative alignment of both is essential. The different states of misalignment of the Shack–Hartmann sensor are divided into groups and are treated theoretically and experimentally. Their effect on the accuracy of wave-front measurements is evaluated. In addition, a practical method for proper alignment of the Shack–Hartmann sensor is proposed.

© 1998 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 323–349.
  2. W. H. Southwell, “Wave front estimation from wave front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  3. G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wave front sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
    [CrossRef]
  4. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  5. C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensors,” Opt. Rev. 2, 217–220 (1995).
    [CrossRef]
  6. C. Witthoft, “Wave front sensor noise reduction and dynamic range expansion by means of optical image intesification,” Opt. Eng. 29, 1233–1238 (1990).
    [CrossRef]
  7. H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
    [CrossRef]

1995 (1)

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensors,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

1994 (2)

G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wave front sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

1990 (1)

C. Witthoft, “Wave front sensor noise reduction and dynamic range expansion by means of optical image intesification,” Opt. Eng. 29, 1233–1238 (1990).
[CrossRef]

1980 (1)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Cao, G.

G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wave front sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

Dainty, J. C.

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensors,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Falkenstörfer, O.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Lindlein, N.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 323–349.

Schwider, J.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Sickinger, H.

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Solomon, C. J.

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensors,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Southwell, W. H.

Witthoft, C.

C. Witthoft, “Wave front sensor noise reduction and dynamic range expansion by means of optical image intesification,” Opt. Eng. 29, 1233–1238 (1990).
[CrossRef]

Wooder, N. J.

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensors,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Yu, X.

G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wave front sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (3)

G. Cao, X. Yu, “Accuracy analysis of a Hartmann–Shack wave front sensor operated with a faint object,” Opt. Eng. 33, 2331–2335 (1994).
[CrossRef]

C. Witthoft, “Wave front sensor noise reduction and dynamic range expansion by means of optical image intesification,” Opt. Eng. 29, 1233–1238 (1990).
[CrossRef]

H. Sickinger, O. Falkenstörfer, N. Lindlein, J. Schwider, “Characterization of microlenses using a phase-shifting shearing interferometer,” Opt. Eng. 33, 2680–2686 (1994).
[CrossRef]

Opt. Rev. (1)

C. J. Solomon, J. C. Dainty, N. J. Wooder, “Bayesian estimation of atmospherically distorted wave fronts using Shack–Hartmann sensors,” Opt. Rev. 2, 217–220 (1995).
[CrossRef]

Proc. IEEE (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other (1)

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp. 323–349.

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Figures (9)

Fig. 1
Fig. 1

Principle of the Shack–Hartmann sensor.

Fig. 2
Fig. 2

Coordinate system of the Shack–Hartmann sensor.

Fig. 3
Fig. 3

Axial displacement of the CCD plane by δz > 0 amplifying the original wave aberrations.

Fig. 4
Fig. 4

Effect of misalignment type (c).

Fig. 5
Fig. 5

Misalignment type (d), mean fit error Γ.

Fig. 6
Fig. 6

Misalignment type (d), PV value of the polynomial.

Fig. 7
Fig. 7

Misalignment (d), plot of polynomial (γ = 2.667̊), PV = 2.669λ.

Fig. 8
Fig. 8

Sum of the gray-scale values of the rows of the CCD frame at the correct γ alignment.

Fig. 9
Fig. 9

Sum of the gray-scale values of the rows of the CCD frame at a misalignment of γ = 0.667̊.

Equations (28)

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

W pq = x y W pq = 1 f × Δ x Δ y pq .
W xy = i = 0 G j = 0 j i   a ij x j y i - j ,
x j y i - j P k x ,   y ,
W x pq W x m ,     W y pq W y m .
G a a j = 0 ,
G a = r = 1 M W xy x r ,   y r x - W x r 2 + W xy x r ,   y r y - W y r 2
A × a = b ,
A mn = r = 1 M x   P m x r ,   y r × x   P n x r ,   y r + y P m x r ,   y r × y   P n x r ,   y r ,
b n = r = 1 M W x r × x   P n x r ,   y r + W y r × y   P n x r ,   y r .
x   W y = y   W x .
r r = r + δ r ,
r r = D α × D β × D γ × r .
Δ r W δ x , δ y = Δ x W + δ x Δ y W + δ y .
W δ x , δ y x = W x + δ x f , W δ x , δ y y = W y + δ y f ,
W δ x , δ y = W + 1 f × x δ x + y δ y ,
f f = f + δ z
W δ z = 1 + δ z f × W .
W β , x = x f × 1 cos   β - 1 + 1 cos   β × W x ,
W β , y = W y .
x   W β , y - y   W β , x = δ x W y × 1 - 1 cos   β - x W y × β 2 2 0   if   β 0 .
Δ r W = r W - r rp .
Δ r W γ = r W γ - r rp = cos   γ sin   γ - sin   γ cos   γ × r W - r rp = x rp + Δ x W × cos   γ - y rp + Δ y W × sin   γ - x rp x rp + Δ x W × sin   γ + y rp + Δ y W × cos   γ - y rp ,
W γ , x = x f + W x × cos   γ - y f + W y × sin   γ - x y ,
W γ , y = x f + W x × sin   γ + y f + W y × cos   γ - y f .
x   W γ ,   y - y   W γ , x = 1   sin   γ × 2 f + 2 W x 2 + 2 W y 2 0   if   γ 0 .
Γ = 1 M × l = 1 M W xy x l ,   y l x - W x l 2 + W xy x l ,   y l y - W y l 2 1 / 2 ,
W plane , β x = x f × 1 cos β - 1 + 1 cos β × W plane x x f × 1 cos β - 1 Δ x = x × 1 cos β - 1 Δ x = Δ x min   β max = arccos x max Δ x min + x max .
Δ y W γ = x rp + Δ x W × sin   γ + y rp + Δ y W × cos   γ - y rp x rp × γ + y rp × 1 - y rp = x rp × γ ,   γ min = Δ y W γ x rp , max = 0.11   μ m 4.4   mm = 2.5 × 10 - 5 rad .

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