Abstract

The feasibility of using null screens for testing the segments of a parabolic segmented telescope mirror for the Large Millimeter Telescope (LMT) is analyzed. An algorithm for designing the null screen for testing the off-axis segments of conic surfaces is described. Actual screen designs for the different classes of segments of the LMT are presented. The sensitivity of the test and the required accuracies for the fabrication and positioning of the screen are analyzed. A measuring accuracy of ∼12 µm in surface sagitta is within the reach of the technique.

© 2000 Optical Society of America

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References

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  1. Instituto Nacional de Astrofisica, Optica y Electrónica, “The Large Millimeter Telescope Homepage,” http://binizaa.inaoep.mx/ ; mail address, Apdo. Postal 51 y 216, 72000 Puebla, Pue., Mexico.
  2. L. Olmi, “The Millimeter Telescope Project: overview and optical design,” http://binizaa.inaoep.mx/pub/014/node4.html ; mail address, Five College Radio Astronomy Observatory, University of Massachusetts, Amherst, Mass. 10020.
  3. By local asphericity I mean the departure of the segment surface from the shape of a sphere whose vertex is coincident with the central point of the segment; its radius of curvature is chosen from the best fit (in the continuum least-squares sense) according to the calculation of Cardona-Núñez et al.,9 with k′ = 0. The local asphericity is a better measure of the departures of the segment surface from a spherical shape.
  4. A. B. Meinel, M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 514–518 (1989).
    [CrossRef]
  5. R. Díaz-Uribe, M. Campos-García, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677 (2000).
    [CrossRef]
  6. By the parent surface I mean the whole surface from which a section is obtained. When all the segments are placed together, the composite surface is the parent surface.
  7. G. W. Hopkins, R. N. Shagam, “Null Ronchi gratings from spot diagrams,” Appl. Opt. 16, 2602–2603 (1977).
    [CrossRef] [PubMed]
  8. A. Cordero-Davila, A. Cornejo-Rodriguez, O. Cardona-Nunez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt. 31, 2370–2376 (1992).
    [CrossRef] [PubMed]
  9. O. Cardona-Nunez, A. Cornejo-Rodriguez, R. Diaz-Uribe, A. Cordero-Davila, J. Pedraza-Contreras, “Conic that best fits an off-axis conic section,” Appl. Opt. 25, 3585–3588 (1986).
    [CrossRef] [PubMed]
  10. C. Menchaca, D. Malacara, “Directional curvatures in a conic mirror,” Appl. Opt. 23, 3258–3259 (1984).
    [CrossRef] [PubMed]
  11. J. C. Wyant, “Interferometric testing of aspheric surfaces,” in Interferometric Metrology (Critical Reviews), N. A. Massie, ed., Proc. SPIE816, 19–39 (1987).
    [CrossRef]
  12. A. Cox, Photographic Optics (Focal Press, London, 1971), pp. 204–205.
  13. R. Diaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vision Sci. 76, 40–49 (1999).
    [CrossRef]
  14. Note that we follow the same notation as that used in Section 3. Again, this is a reverse ray trace; however, it does not affect to the evaluation of the normal to the surface.

2000 (1)

1999 (1)

R. Diaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vision Sci. 76, 40–49 (1999).
[CrossRef]

1992 (1)

1989 (1)

A. B. Meinel, M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 514–518 (1989).
[CrossRef]

1986 (1)

1984 (1)

1977 (1)

Campos-García, M.

Cardona-Nunez, O.

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Cox, A.

A. Cox, Photographic Optics (Focal Press, London, 1971), pp. 204–205.

Diaz-Uribe, R.

Díaz-Uribe, R.

Granados-Agustin, F.

R. Diaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vision Sci. 76, 40–49 (1999).
[CrossRef]

Hopkins, G. W.

Malacara, D.

Meinel, A. B.

A. B. Meinel, M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 514–518 (1989).
[CrossRef]

Meinel, M. P.

A. B. Meinel, M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 514–518 (1989).
[CrossRef]

Menchaca, C.

Pedraza-Contreras, J.

Shagam, R. N.

Wyant, J. C.

J. C. Wyant, “Interferometric testing of aspheric surfaces,” in Interferometric Metrology (Critical Reviews), N. A. Massie, ed., Proc. SPIE816, 19–39 (1987).
[CrossRef]

Appl. Opt. (5)

Opt. Eng. (1)

A. B. Meinel, M. P. Meinel, “Optical testing of off-axis parabolic segments without auxiliary optical elements,” Opt. Eng. 28, 514–518 (1989).
[CrossRef]

Optom. Vision Sci. (1)

R. Diaz-Uribe, F. Granados-Agustin, “Corneal shape evaluation by using laser keratopography,” Optom. Vision Sci. 76, 40–49 (1999).
[CrossRef]

Other (7)

Note that we follow the same notation as that used in Section 3. Again, this is a reverse ray trace; however, it does not affect to the evaluation of the normal to the surface.

Instituto Nacional de Astrofisica, Optica y Electrónica, “The Large Millimeter Telescope Homepage,” http://binizaa.inaoep.mx/ ; mail address, Apdo. Postal 51 y 216, 72000 Puebla, Pue., Mexico.

L. Olmi, “The Millimeter Telescope Project: overview and optical design,” http://binizaa.inaoep.mx/pub/014/node4.html ; mail address, Five College Radio Astronomy Observatory, University of Massachusetts, Amherst, Mass. 10020.

By local asphericity I mean the departure of the segment surface from the shape of a sphere whose vertex is coincident with the central point of the segment; its radius of curvature is chosen from the best fit (in the continuum least-squares sense) according to the calculation of Cardona-Núñez et al.,9 with k′ = 0. The local asphericity is a better measure of the departures of the segment surface from a spherical shape.

By the parent surface I mean the whole surface from which a section is obtained. When all the segments are placed together, the composite surface is the parent surface.

J. C. Wyant, “Interferometric testing of aspheric surfaces,” in Interferometric Metrology (Critical Reviews), N. A. Massie, ed., Proc. SPIE816, 19–39 (1987).
[CrossRef]

A. Cox, Photographic Optics (Focal Press, London, 1971), pp. 204–205.

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Figures (14)

Fig. 1
Fig. 1

Geometry of the array of 126 segments for the LMT. Shading has been added to identify segments of the same class. Offset x c (the distance between the array and individual segment centers) determines the class. There are 15 classes. Classes A, B, C, E, F, H, K, L, and O can have 6 members and classes D, G, I, J, M, and N can have 12 members.

Fig. 2
Fig. 2

Optical layout for testing the segments with null screens. I and R are the unit vectors along the incident and the reflected rays, respectively. They are oriented as the design procedure goes on; they are reversed to the actual ray propagation during the testing.

Fig. 3
Fig. 3

Plot of the actual X (open circles) and Y (open squares) lengths of the null screen designed as a function of screen position a for testing class A segments of the LMT. The vertical dashed lines denote the positions of the s.c.c. and of the t.c.c. as calculated for the center of the segment. As in the paraxial calculation, the behavior is linear, but the zero value does not lie at the centers of curvature [see Eqs. (11) and (12)].

Fig. 4
Fig. 4

Null screens designed for testing the segment classes of the LMT; (a) is for class A segments, (b) is for class B segments, and so on. The small circles locate the crossing points. Note the scale change among the plots.

Fig. 5
Fig. 5

Plot of screen–surface distance a of the null screens for testing the segments of the LMT, as a function of segment offset distance x c . Each design was done in such a way that maximum sensitivity was attained with a smaller screen–surface distance, avoiding the caustic zone. For reference, the sagittal and tangential radii of curvature (s.r.c. and t.r.c., respectively) for each segment at its center are also plotted.

Fig. 6
Fig. 6

Plot of actual X and Y lengths D sx and D sy , respectively, of the null screens for testing the segments of the LMT as functions of segment offset x c .

Fig. 7
Fig. 7

Simulated image on the CCD for the test of LMT class A segments. Deformation of the surface is introduced by a small change on the conic constant of the surface. Design conic constant, k = -1. Modified conic constants: (a) k′ = -1.001; (b) k′ = -1.01; (c) k′ = -1.05; (d) k′ = -1.1; (e) k′ = -1.5.

Fig. 8
Fig. 8

Differences dx and dy between the coordinates of the simulated crossing points and the ideal positions on the CCD for the test of LMT class A segments: (a) k′ = -1.001, (b) k′ = -1.005, (c) k′ = -1.01, (d) k′ = -1.05, (e) k′ = -1.1. For comparison, a gray square with a size equal to one pixel on the CCD (9 µm × 9 µm), is also shown.

Fig. 9
Fig. 9

Log–log plot of the average distances 〈ρ〉 between the image points for simulated testing of a deformed surface and the ideal points as functions of the conic constant variation k - k′ = -k′ - 1 (thicker solid curve). The maximum values for dx (thinner solid curve) and dy (short-dashed curve) are also plotted.

Fig. 10
Fig. 10

Plot of the conic constant difference limiting value Δk min that can be sensed versus the offsets of the various segment classes for the LMT.

Fig. 11
Fig. 11

Maximum sagittal differences Δz max between the deformed and a perfect surface along a 5 m × 5 m square area for each segment, corresponding to the Δk min values of Fig. 10 versus offset x c of the segment classes for the LMT.

Fig. 12
Fig. 12

Plots of average distances 〈ρ〉 for class A segments as functions of screen position a. Modified conic constants k′ are shown. For reference, the vertical lines at the s.c.c. and the t.c.c. are shown, as well as the 9-µm pixel size (horizontal line).

Fig. 13
Fig. 13

Plotting accuracies ΔX and ΔY needed for fabrication of null screens as functions of offset x c of the segment classes for the LMT. As the accuracy depends on the point position on the screen, the maximum and minimum values are plotted. For reference, the attainable resolutions of commercial printers are plotted (horizontal dotted–dashed lines).

Fig. 14
Fig. 14

Plot of the position change of the image grid points for screen defocusing (da increments in distance a) as a function of offset x c of the segment classes for the LMT. When the screen is moved along the z axis a distance da away from the design position, the image of each crossing point on the CCD changes its coordinates in dx and dy amounts along each transversal axis. The average distance 〈Δρ〉 and the maximum values are plotted for da = 0.1 mm and a = 1.0 mm.

Tables (1)

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Table 1 Offset and Asphericity of Each Segment According To Its Class

Equations (35)

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I=-x0, -y0, a-bx02+y02+a-b21/2.
x=z-ab-a x0,
y=z-ab-a y0,
zx, y=DB+B2-AD1/2,
A=c1+k cos2 θ,
B=11+k sin2 θ1/2-ck sin θ cos θ x,
D=c1+k sin2 θx2+cy2,
tan θ=cxc1-k+1c2xc21/2,
y=-β±β2-4αγ1/22α,
α=Aa-b2+Ex02y02+c-2S a-bx0y02,
β=2y0aSx0+a-bc/E-aA,
γ=a2A-2ac/E.
S=ck sin θ cos θ,
E=c1+k sin2 θ.
x=x0y0 y.
R=I-2I·NN,
N=-zx, -zy, 11+zx2+zy21/2.
zx=DT+BDS-BT+B2-AD/22ExB+S2S,
zy=BT+B2-AD/22cyB+S2S,
X=x+a-zx0zx2-zy2-1+2zxy0zy+a-ba-bzx2+zy2+1-2x0zx+y0zy,
Y=y+a-zy0zy2-zx2-1+2zyx0zx+a-ba-bzx2+zy2+1-2x0zx+y0zy.
x0=nd-L/2,
y0=md-L/2,
b=1-LD0a,
Ds=2-afD0,
ft=1-kc2xc23/22c,
fs=1-kc2xc21/22c.
ρ=1N2i=1Nj=1Ndxij2+dyij21/2.
Δx=dlxij ΔX.
distortion percent=ξβ×α2+β2αβ,
h>2.44 λb-aAmax.
N·S=0.
z-zi=- NxNzdx+NyNzdy,
R=X-xs, Y-ys, a-zsX-xs2+Y-ys2+a-zs21/2,
N=R-I|R-I|.

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