Abstract

We present calculations with an exact ray trace to determine the dimensions that define one or two Hindle spheres, since the paraxial theory is incongruent for convex hyperboloid mirrors with small f numbers. The equations are generalized to calculate the dimensions of n Hindle spheres, since in this way it is possible to reduce the dimensions of the spheres more. Actual calculations are done for the secondary mirrors of the Benemerita Universidad Autonoma de Puebla and Large Milimetric Telescopes; experimental results are shown for the latter.

© 1999 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley, New York, 1978).
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    [CrossRef] [PubMed]
  3. I. S. Potyemin, A. S. Seregin, “New modification of Hindle scheme for interferometric testing of convex hyperbolical surfaces,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 276–282 (1994).
  4. J. Hindle, “A new test for Cassegrainian and Gregorian secondary mirrors,” Mon. Not. R. Astron. Soc. 91, 592–595 (1931).
  5. A. Cordero-Dávila, E. Luna-Aguilar, S. Vázquez-Montiel, S. Zárate-Vázquez, M. E. Percino-Zacarias, “Ronchi test using a square grid,” Appl. Opt. 37, 672–675 (1998).
    [CrossRef]
  6. A. Cornejo, D. Malacara, “Ronchi test of aspherical surfaces, analysis, and accuracy,” Appl. Opt. 9, 1897–1901 (1970).
    [PubMed]
  7. Y. M. Liu, G. N. Lawrence, C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27, 4504–4513 (1988).
    [CrossRef] [PubMed]
  8. M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
    [CrossRef]

1998

1993

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

1988

1974

1970

1931

J. Hindle, “A new test for Cassegrainian and Gregorian secondary mirrors,” Mon. Not. R. Astron. Soc. 91, 592–595 (1931).

Cordero-Dávila, A.

Cornejo, A.

Hindle, J.

J. Hindle, “A new test for Cassegrainian and Gregorian secondary mirrors,” Mon. Not. R. Astron. Soc. 91, 592–595 (1931).

Koliopoulos, C. L.

Lawrence, G. N.

Liu, Y. M.

Luna-Aguilar, E.

Malacara, D.

Mazzoni, A.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Melozzi, M.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Noble, R.

Percino-Zacarias, M. E.

Pezzati, L.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Potyemin, I. S.

I. S. Potyemin, A. S. Seregin, “New modification of Hindle scheme for interferometric testing of convex hyperbolical surfaces,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 276–282 (1994).

Seregin, A. S.

I. S. Potyemin, A. S. Seregin, “New modification of Hindle scheme for interferometric testing of convex hyperbolical surfaces,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 276–282 (1994).

Vázquez-Montiel, S.

Zárate-Vázquez, S.

Appl. Opt.

Mon. Not. R. Astron. Soc.

J. Hindle, “A new test for Cassegrainian and Gregorian secondary mirrors,” Mon. Not. R. Astron. Soc. 91, 592–595 (1931).

Opt. Eng.

M. Melozzi, L. Pezzati, A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32, 1073–1079 (1993).
[CrossRef]

Other

D. Malacara, Optical Shop Testing (Wiley, New York, 1978).

I. S. Potyemin, A. S. Seregin, “New modification of Hindle scheme for interferometric testing of convex hyperbolical surfaces,” in Interferometry ’94: New Techniques and Analysis in Optical Measurements, M. Kujawinska, K. Patorski, eds., Proc. SPIE2340, 276–282 (1994).

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Figures (4)

Fig. 1
Fig. 1

Hindle arrangement for testing a convex hyperboloid.

Fig. 2
Fig. 2

Arrangement of two Hindle spheres for a testing a convex hyperboloid.

Fig. 3
Fig. 3

Experimental arrangement of two Hindle spheres.

Fig. 4
Fig. 4

Ronchigrams of the two-Hindle-sphere setup: (a) Ronchi grid parallel to the x axis; (b) Ronchi grid parallel to the y axis. Ronchigrams of the one-Hindle-sphere setup: (c) Ronchi grid parallel to the x axis; (d) Ronchi grid parallel to the y axis.

Tables (7)

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Table 1 BUAP Secondary Mirror Parameters

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Table 2 Hindle-Sphere Parameters when Paraxial and Exact Ray Tracing for a BUAP Secondary Mirror are Used

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Table 3 LMT Secondary Mirror Parameters

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Table 4 Hindle-Sphere Parameters when Paraxial Exact Ray Tracing for LMT Secondary Mirrors are Used

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Table 5 Two Hindle-Sphere Parameters Obtained by a Paraxial Ray for the Convex Mirror of the BUAP

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Table 6 Parameters for Two Hindle Spheres when First and Second Criteria for the Convex Mirror of the BUAP Are Applied

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Table 7 n-Hindle-Sphere Parameters when Exact Ray Tracing for LMT Secondary Mirrors Are Used

Equations (17)

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yAzA-f1=YBZB-f1,
yAzA-f2=yCzC-f2.
yA=yCYBf1-f2yCZB-f1-YBzC-f2,
zA=yCf2ZB-f1-YBf1zC-f2yCZB-f1-YBzC-f2.
RH=YBf1-f2zC-f22+yC21/2yCZB-f1-YBzC-f2.
YD=YBYBf1-f2zC-f22+yC21/2ZB-f22+YB21/2yCZB-f1-YBzC-f2.
ZB=cYB21+1-k+1c2YB21/2,
YD1=YD2.
YBYBf1-f2zC1-f22+yC121/2f2-ZB2+YB21/2yC1ZB-f1-YBzC1-f2=yC1YBf1-f2zC2-f22+yC221/2f2-zC12+yC121/2yC2ZB-f1-YBzC2-f2.
yC12+yC1YBzC1-f2f2-zC22+yC221/2f2-ZB2+YB21/2YByC2ZB-f1-YBzC1-f2-ZB-f1f2-zC22+yC22f2-ZB2+YB21/2+YBzC1-f22yC2ZB-f1-YBzC1-f2YByC2ZB-f1-YBzC1-f2-ZB-f1f2-zC2)2+yC22f2-ZB2+YB21/2=0.
zC1=cyC121+1-k+1c2yC121/2.
yC1=YB+yC22,
yCj=yCj-1-YB-yCnn-1,   j=2, 3,, n-1.
yAj=yCjYBf1-f2yCjZB-f1-YBzCj-f2,
zAj=yCjf2ZB-f1-YBf1zCj-f2yCjZB-f1-YBzCj-f2,
RHj=f2-zAj2+yAj21/2,
YDj=yCjRHjzCj-f22+yCj1/2.

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