Abstract

In the classical Hartmann test the wave front is obtained by integration of the transverse aberrations, joining the sampled points by small straight segments, in the so-called Newton integration. This integration is performed along straight lines joining the holes on the Hartmann screen. We propose a modification of this procedure, considering the cells of four holes of the Hartmann screen to fit a small second-power wave front recovering each square. This procedure has some important advantages, as described here.

© 2005 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. I. Ghozeil, J. E. Simmons, “Screen test for large mirrors,” Appl. Opt. 13, 1773–1777 (1974).
    [CrossRef] [PubMed]
  3. J. D. Mansell, J. Hennawi, E. K. Gustafson, M. M. Fejer, R. L. Byer, D. Clubley, S. Yoshida, D. H. Reitze, “Evaluating the effect of transmissive optic thermal lensing on laser beam quality with a Shack–Hartmann wave-front sensor,” Appl. Opt. 40, 366–374 (2001).
    [CrossRef]
  4. D. Malacara, “Testing and centering by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
    [CrossRef]

2001 (1)

1992 (1)

D. Malacara, “Testing and centering by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

1974 (1)

Byer, R. L.

Clubley, D.

Fejer, M. M.

Ghozeil, I.

I. Ghozeil, J. E. Simmons, “Screen test for large mirrors,” Appl. Opt. 13, 1773–1777 (1974).
[CrossRef] [PubMed]

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

Gustafson, E. K.

Hennawi, J.

Malacara, D.

D. Malacara, “Testing and centering by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

Mansell, J. D.

Reitze, D. H.

Simmons, J. E.

Yoshida, S.

Appl. Opt. (2)

Opt. Eng. (1)

D. Malacara, “Testing and centering by means of a Hartmann test with four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

Other (1)

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 (11)

Fig. 1
Fig. 1

Classical Hartmann test.

Fig. 2
Fig. 2

(a) Hartmann screen; (b) Hartmann plate in the classical test.

Fig. 3
Fig. 3

Example of a trapezoidal integration of transverse aberration measurements.

Fig. 4
Fig. 4

Unit cell in a Hartmann screen.

Fig. 5
Fig. 5

Arrangements of the four Hartmann spots from a unit cell for some aberrations: (a) astigmatism at 0° or 90°; (b) astigmatism at an angle ±45°; (c) astigmatism at an angle of θ, (d) coma in the y direction.

Fig. 6
Fig. 6

Some variables used for a set of four measurements.

Fig. 7
Fig. 7

Wave front produced by an aspheric wave front with aspheric deformation terms.

Fig. 8
Fig. 8

Calculated transverse aberrations for the aspheric wave front.

Fig. 9
Fig. 9

(a) Ray intersections at the pupil (Hartmann screen); (b) spot diagram (Hartmann pattern) produced by the aspheric wave front.

Fig. 10
Fig. 10

Wave-front profile calculated with classical trapezoidal integration and the difference between the original wave front.

Fig. 11
Fig. 11

Wave-front profile calculated with the new method and the difference between the original wave front.

Equations (40)

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W ( x , y ) x = T A x ( x , y ) r = θ x ( x , y ) ,
W ( x , y ) y = T A y ( x , y ) r = θ y ( x , y ) .
S = 1 2 K [ K η x ( n , m ) n + K η y ( n , m ) m ] ,
T A x ( n , m ) = η x ( n , m ) n S , T A y ( n , m ) = η y ( n , m ) m S .
W ( n , m ) = d 2 r i = 1 n [ T A x ( i 1 , m ) + T A x ( i , m ) ]
W ( n , m ) = d 2 r j = 1 m [ T A y ( n , j 1 ) + T A y ( n , j ) ]
W ( x n , y m ) = W ( x n 1 , y m ) + d 2 r [ T A x ( x n 1 , y m ) + T A x ( x n , y m ) ] ,
W ( x n , y m ) = W ( x n , y m 1 ) + d 2 r [ T A y ( x n , y m 1 ) + T A y ( x n , y m ) ] ,
T A x ( x , y m ) = T A x ( x n 1 , y m ) + [ T A x ( x n , y m ) T A x ( x n 1 , y m ) d ] x ,
T A y ( x n , y ) = T A y ( x n , y m 1 ) + [ T A y ( x n , y m ) T A y ( x n , y m 1 ) d ] y .
W ( x , y m ) = W ( x n 1 , y m ) + [ T A x ( x n 1 , y m ) r ] x + [ T A x ( x n , y m ) T A x ( x n 1 , y m ) 2 r d ] x 2 ,
W ( x n , y ) = W ( x n , y m 1 ) + [ T A y ( x n , y m 1 ) r ] y + [ T A y ( x n , y m ) T A y ( x n , y m 1 ) 2 r d ] y 2 = W ( x n , y m 1 ) + [ T A y ( x n , y m 1 ) r ] y + c 2 y 2 ,
W n m ( x , y ) = A n , m + B n , m x + C n , m y + D n , m ( x 2 + y 2 ) + E n , m ( x 2 y 2 ) + F n , m x y + G n , m ( x 2 + y 2 d 2 ) y + H n , m ( x 2 + y 2 d 2 ) x ,
W ( x , y ) x = T A x r = ϕ x = B + 2 D x + 2 E x + F y + 2 G x y + H ( 3 x 2 + y 2 d 2 ) ,
W ( x , y ) y = T A y r = ϕ y = C + 2 D y 2 E y + F x + G ( x 2 + 3 y 2 d 2 ) + 2 Hxy ,
T A x 0 ( n , m ) = B n , m r = T A x α + T A x β + T A x γ + T A x δ 4 ,
T A y 0 ( n , m ) = C n , m r = T A x α + T A x β + T A x γ + T A x δ 4 ,
ϕ x α = T A x α T A x 0 r = ( D + E + F 2 ) d + G 2 d 2 , ϕ y α = T A y α T A y 0 r = ( D E + F 2 ) d + H 2 d 2 , ϕ x β = T A x β T A x 0 r = ( D + E F 2 ) d G 2 d 2 , ϕ y β = T A y β T A y 0 r = ( D E F 2 ) d H 2 d 2 , ϕ x γ = T A x γ T A x 0 r = ( D + E + F 2 ) d + G 2 d 2 , ϕ y γ = T A y γ T A y 0 r = ( D E + F 2 ) d + H 2 d 2 , ϕ x δ = T A x δ T A x 0 r = ( D + E F 2 ) d G 2 d 2 , ϕ y δ = T A y δ T A y 0 r = ( D E F 2 ) d H 2 d 2 .
D n , m = ( T A x α T A x β ) ( T A x γ T A x δ ) + ( T A y α + T A y β ) ( T A y γ + T A y δ ) 8 r d , E n , m = ( T A x α T A x β ) ( T A x γ T A x δ ) ( T A y α + T A y β ) + ( T A y γ + T A y δ ) 8 r d , F n , m = ( T A x α + T A x β ) ( T A x γ T A x δ ) 2 r d , G n , m = ( T A x α T A x β ) + ( T A x γ T A x δ ) 2 r d 2 , H n , m = ( T A y α T A y β ) + ( T A y γ T A y δ ) 2 r d 2 .
ϕ x α = T A x α T A x 0 r = ( D + E + F 2 ) d , ϕ y α = T A y α T A y 0 r = ( D E + F 2 ) d , ϕ x β = T A x β T A x 0 r = ( D + E F 2 ) d , ϕ y β = T A y β T A y 0 r = ( D E F 2 ) d , ϕ x γ = T A x γ T A x 0 r = ( D + E + F 2 ) d , ϕ y γ = T A y γ T A y 0 r = ( D E + F 2 ) d , ϕ x δ = T A x δ T A x 0 r = ( D + E F 2 ) d , ϕ y δ = T A y δ T A y 0 r = ( D E F 2 ) d .
D = T A x β T A y β T A x δ + T A y δ + 2 ( T A x γ + T A y γ T A x 0 T A y 0 ) 8 r d , E = T A x β + T A y β + T A x γ T A y γ T A x δ T A y δ T A x 0 + T A y 0 ) 6 r d , F = 2 ( T A x γ + T A y γ T A x 0 + T A y 0 ) T A x β + T A y β + T A x δ T A y δ 4 r d .
D = T A x γ T A y γ T A x α T A y α 2 ( T A x δ T A y δ T A x 0 + T A y 0 ) 8 r d , E = T A x γ T A y γ T A x α + T A y α T A x δ T A y δ + T A x 0 + T A y 0 ) 6 r d , F = T A x γ + T A y γ T A x α T A y α 2 ( T A x δ T A y δ T A x 0 + T A y 0 ) 4 r d .
D = T A x β T A y β T A x δ + T A y δ 2 ( T A x α + T A y α T A x 0 T A y 0 ) 8 r d , E = T A x β + T A y β T A x α + T A y α T A x δ T A y δ + T A x 0 + T A y 0 ) 6 r d , F = 2 ( T A x α + T A y α T A x 0 T A y 0 ) + T A x β T A y β T A x δ + T A y δ 4 r d .
D = T A x γ + T A y γ T A x α T A y α + 2 ( T A x β T A y β T A x 0 + T A y 0 ) 8 r d , E = T A x β + T A y β T A x α + T A y α + T A x γ T A y γ T A x 0 T A y 0 ) 6 r d , F = T A x γ + T A y γ T A x α T A y α 2 ( T A x β T A y β T A x 0 + T A y 0 ) 4 r d .
W α = ( B + C ) 2 d + ( 2 D + F ) 4 d + ( G + H ) 4 d .
W β = ( B C ) 2 + D 2 d 2 F 4 d 2 + ( G H ) 4 d 3 .
W γ = ( B + C ) 2 d + D 2 d 2 + F 4 d 2 + ( G + H ) 4 d 3 ,
W δ = ( B C ) 2 d + ( 2 D F ) 2 d 2 ( G H ) 4 d 3 ,
A n , m = 1 2 [ W β ( n 1 , m ) + W γ ( n 1 , m ) ] 1 2 [ W α ( n , m ) + W β ( n , m ) ]
A n , m = 1 2 [ W δ ( n , m 1 ) + W γ ( n , m 1 ) ] 1 2 [ W α ( n , m ) + W δ ( n , m ) ]
W β = ( B C ) 2 d + ( 2 D F ) 4 d 2 , W γ = ( B + C ) 2 d + ( 2 D + F ) 4 d 2 , W δ = ( B C ) 2 d + ( 2 D F ) 4 d 2 .
W α = ( B + C ) 2 d + ( 2 D + F ) 4 d 2 , W γ = ( B + C ) 2 d + ( 2 D + F ) 4 d 2 , W δ = ( B C ) 2 d + ( 2 D F ) 4 d 2 .
W α = ( B + C ) 2 d + ( 2 D + F ) 4 d 2 , W β = ( B C ) 2 d + ( 2 D F ) 4 d 2 , W γ = ( B C ) 2 d + ( 2 D F ) 4 d 2 .
W α = ( B + C ) 2 d + ( 2 D + F ) 4 d 2 , W β = ( B C ) 2 d + ( 2 D F ) 4 d 2 , W γ = ( B + C ) 2 d + ( 2 D + F ) 4 d 2 .
W ( x , y ) = D ( x 2 + y 2 ) + E ( x 2 + y 2 ) + Fxy ,
W ( r , θ ) = ( D + E cos 2 θ + F cos θ sin θ ) r 2 = ( D + E cos 2 θ + F sin 2 θ ) r 2 .
c r = 2 W r 2 = 2 D + 2 E cos 2 θ + F sin 2 θ ,
C r = 2 D + ( 4 E 2 + F 2 ) 1 / 2 cos 2 ( θ α ) = 2 D + ( 4 E 2 + F 2 ) 1 / 2 2 ( 4 E 2 + F 2 ) 1 / 2 sin 2 ( θ α ) ,
tan 2 α = F 2 E .
c r = 2 ( 4 E 2 + F 2 ) 1 / 2 .

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