Abstract

In order to introduce many more evaluation points during the Hartmann test, the scanning of the screen across the pupil is proposed; after each step of the scan a different image of the bright spots is obtained. Basic ideas about how to design radial and square screens for the scanning are presented. Radial screens are scanned by rotation, whereas for square screens a linear inclined scan is enough to introduce many more evaluation points along two independent directions. For square screens it is experimentally shown that the lateral resolution of the test is improved.

© 2009 OSA

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References

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  1. D. Malacara-Doblado and E. Ghozeil, “Hartmann, Hartmann-Shack and other screen tests,” in Optical Shop Testing, Third Edition, Daniel Malacara, ed., (John Wiley and Sons, 2007), Chapter10.
  2. A. Morales and D. Malacara, “Geometrical parameters in the Hartmann test of aspherical mirrors,” Appl. Opt. 22(24), 3957–3959 (1983).
    [CrossRef] [PubMed]
  3. V. I. Moreno-Oliva, M. Campos-García, R. Bolado-Gómez, and R. Díaz-Uribe, “Point shifting in the optical testing of fast aspheric concave surfaces by a cylindrical screen,” Appl. Opt. 47(5), 644–651 (2008).
    [CrossRef] [PubMed]
  4. D. Liu, H. Huang, B. Ren, A. Zeng, Y. Yan, and X. Wang, “Scanning Hartmann test method and its application to lens aberration measurement,” Chin. Opt. Lett. 4, 725–728 (2006).
  5. R. Díaz-Uribe, “Medium-precision null-screen testing of off-axis parabolic mirrors for segmented primary telescope optics: the large millimeter telescope,” Appl. Opt. 39(16), 2790–2804 (2000).
    [CrossRef]
  6. M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48(5), 1008–1015 (2009).
    [CrossRef] [PubMed]

2009 (1)

2008 (1)

2006 (1)

2000 (1)

1983 (1)

Avendaño-Alejo, M.

Bolado-Gómez, R.

Campos-García, M.

Díaz-Uribe, R.

Huang, H.

Liu, D.

Malacara, D.

Morales, A.

Moreno-Oliva, V. I.

Ren, B.

Wang, X.

Yan, Y.

Zeng, A.

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Figures (15)

Fig. 1.
Fig. 1.

Design of the spiral screen. a) The basic set of points is defined; in this case n = 4, m = 5, θ = 90°. b) Additional points are included k = 5. c) Final positions for the holes on the screen. d) Rotation of the screen in steps of ∆α = 3.6° allows testing of the surface at T = 3000 points.

Fig. 2.
Fig. 2.

Optimum hole diameter dh as a function of the distance L, for K = 0, λ = 500 nm, r = 1/c = 600 mm, D = 150mm, according to ref [3] (see Eq. (1).

Fig. 3.
Fig. 3.

Spiral screen as designed in Fig. 1 (c), with finite hole size. The superposition at the center is evident. Units are mm.

Fig. 4.
Fig. 4.

Spiral screen without overlapping; a more uniform sampling is made.

Fig. 5.
Fig. 5.

Sampled points (blue dots) with the screen in Fig. 4 [n = m = 20], for different numbers of rotation movements and images: a) t = 4, b) t = 8, c) t = 12, d) t = 17, e) t = 32; each angular step is γ = 2π/ t.

Fig. 6.
Fig. 6.

Sampled points with the screen in Fig. 4 rotated around an axis 3mm to the right of the center of the screen; for this case, n = m = 20, t = 10, the angular step is γ = 2π / t. The sampling is more uniform even at the center, but the points are distributed more randomly.

Fig. 7.
Fig. 7.

Parameters of the scanning of a classical square Hartmann test. Solid dots (red) are the center points for the holes in a classical square Hartmann screen; hollow dots (blue) are the new points added by the scan. In this case: b = 3, s = 8. Each hole at the classical square Hartmann screen adds s-1 new evaluation points at the surface; for instance, point 0 gives points 1, 2, 3, …, 7 after the scan. For this example, k = 2, d 2 = 0.354 c, d 3 = 0.395 c, α 2 = 45°, α 3 = -71.565° ds = 0.395 dsx = 0.375 and dsy = 0.125.

Fig. 8.
Fig. 8.

Examples of additional points added by the scan (hollow blue dots) for the cases a) s = 4, b) s = 5, c) s = 6, d) s = 7, e) s = 8, f) s = 9, g) s = 10, h) s = 11, i) s = 12, j) s = 13, k) s = 14, and l) s = 15. In every case b = 3.

Fig. 9.
Fig. 9.

Design of the Hartmann screen for testing a mirror with scan. The arrow shows the scan direction. Distances are in mm.

Fig. 10.
Fig. 10.

Experimental setup for the scanning Hartmann test of the spherical mirror. Distances are: A = 585 mm, B = 689 mm, and C = 135 mm

Fig. 11.
Fig. 11.

a) Actual screen for the test; it is inclined and mounted on a precision stage for the scan. The test surface is behind the screen. b) Hartmanngram for the starting position of the screen (0.00 mm); the other images are for c) 9.84 mm, d) 19.68, and e) 29.52 mm.

Fig. 12.
Fig. 12.

Evaluated positions of the centroids of the 9 images (in pixels); the blue squares are the centroids obtained for image in Fig. 11.b) at the starting position of the screen. For each position ten images are averaged to reduce the noise.

Fig. 13.
Fig. 13.

Diagram to explain the vectors involved in the calculation of the normal of the test surface.

Fig. 14.
Fig. 14.

Integration paths for evaluating the surface with a) only one image (89 points), and b) with the 9 images (803 points). Distances are in mm.

Fig. 15.
Fig. 15.

Surface differences from the best fit surface. A) Results obtained with 90 points from only one image b) Results obtained for all the 803 points obtained from the 9 images of the scan. All distances are in mm.

Tables (1)

Tables Icon

Table 1. Geometrical parameters for the array of evaluation points generated during the linear scan of a classical square Hartmann screen. For these data b = 3 and c = 1, is used; for all the cases β = 18.435. Angular data are given in degrees.

Equations (29)

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θ=2πn ;
a=D2m
Δa=ak+1=D2m(k+1),
α=θk+1 .
N=nm(k+1) .
s=(k+1)2
Δα=αk+1.
T=N(k+1)2=nm(k+1)3
dh=[1.63λKc3D2+1.33c2L]
c=Dm1,
Nπ(m1)24.
d=cb,
β=tan1(1b) .
d=c(1+b2)1/2 .
ds=ds=c(1+b2)1/2s ;
T=Ns14π(m1)2s.
dsx=dscosβ=cbs ,
dsy=dssinβ=cs.
k=int(sb)
dj=cs [(sjb)2+j2] .
αj=tan1(jsjb)
Ri=V1+H,
Rr=V2+SH,
N=rrrirrri,
zzo=(nxnzdx+nynzdy);
εh212(ba)M,
z(x)=Ax2 +Bx+C,
N(x,z)=(2Ax+B)îk̂,
f(x)=NxNz=2AxB.

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