Abstract

A method of measuring the profile along one diameter of a conic surface is presented. Using some well-known formulas for the conic sections, useful mathematical relations are derived that, together with a simple experimental setup can be used for comparing theoretical and real values of the sagitta of the surface.

© 1985 Optical Society of America

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References

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  1. W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, New York, 1978).
  2. A. Cornejo-Rodriguez, D. Malacara-Hernandez, “Ronchi Test of Aspherical Surfaces; Analysis and Accuracy,” Appl. Opt. 9, 1897 (1970).
  3. D. Malacara-Hernandez, A. Cornejo-Rodriguez, “Null Ronchi Test for Aspherical Surfaces,” Appl. Opt. 13, 1778 (1974).
    [CrossRef]
  4. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972). p. 91.
  5. A. Cornejo-Rodriguez, D. Malacara-Hernandez, “Caustic Coordinates in Platzeck-Gaviola Test for Conic Mirrors,” Appl. Opt. 17, 18 (1978).
    [CrossRef]
  6. J. Pedraza-Contreras, A. Cornejo-Rodriguez, A. Cordero-Davila, “Formulas for Setting the Diamond Tool in the Precision Machining of Conic-Surfaces,” Appl. Opt. 20, 2882 (1981).
    [CrossRef] [PubMed]
  7. B. A. Chunin, S. Kachkin, “Methods of Controlling the Shape of Aspheric Surfaces,” Sec. 1: “Mechanical Methods,” in Generation of Optical Surfaces, K. J. Kumanin, Ed. [Focal Library, London, 1962 (Russian edition)], Chap. 9, pp. 294–300.
  8. T. M. Leushina, “Testing of Aspherical Surfaces on a Goniometer,” Sov. J. Opt. Technol. 45, 544 (1978).
  9. Ref. 7, Section 3: “Aspherometer,” in Generation of Optical Surfaces, K. J. Kumanin, Ed. [The Focal Library, London, 1962 (Russian edition)], pp. 312–325.
  10. R. S. Longhurst, Geometrical and Physical Optics (Wiley, New York, 1964), p. 68.
  11. A. Cornejo-Rodriguez, A. Cordero-Davila, “Measurement of Radii of Curvature of Convex and Concave Surfaces Using a Nodal Bench and He–Ne Laser,” Appl. Opt. 19, 1743 (1980).
    [CrossRef] [PubMed]
  12. A. Cornejo-Rodriquez et al., “Profile Measurement of a Conic Surface Using a Nodal Bench and a He-Ne Laser,” in Conference Digest, Optics in Modern Science Technology, ICO-13, Sapporo, Japan (1984), p. 402.

1981 (1)

1980 (1)

1978 (2)

A. Cornejo-Rodriguez, D. Malacara-Hernandez, “Caustic Coordinates in Platzeck-Gaviola Test for Conic Mirrors,” Appl. Opt. 17, 18 (1978).
[CrossRef]

T. M. Leushina, “Testing of Aspherical Surfaces on a Goniometer,” Sov. J. Opt. Technol. 45, 544 (1978).

1974 (1)

1970 (1)

Chunin, B. A.

B. A. Chunin, S. Kachkin, “Methods of Controlling the Shape of Aspheric Surfaces,” Sec. 1: “Mechanical Methods,” in Generation of Optical Surfaces, K. J. Kumanin, Ed. [Focal Library, London, 1962 (Russian edition)], Chap. 9, pp. 294–300.

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Cornejo-Rodriquez, A.

A. Cornejo-Rodriquez et al., “Profile Measurement of a Conic Surface Using a Nodal Bench and a He-Ne Laser,” in Conference Digest, Optics in Modern Science Technology, ICO-13, Sapporo, Japan (1984), p. 402.

Kachkin, S.

B. A. Chunin, S. Kachkin, “Methods of Controlling the Shape of Aspheric Surfaces,” Sec. 1: “Mechanical Methods,” in Generation of Optical Surfaces, K. J. Kumanin, Ed. [Focal Library, London, 1962 (Russian edition)], Chap. 9, pp. 294–300.

Leushina, T. M.

T. M. Leushina, “Testing of Aspherical Surfaces on a Goniometer,” Sov. J. Opt. Technol. 45, 544 (1978).

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics (Wiley, New York, 1964), p. 68.

Malacara-Hernandez, D.

Pedraza-Contreras, J.

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972). p. 91.

Welford, W. T.

W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, New York, 1978).

Winston, R.

W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, New York, 1978).

Appl. Opt. (5)

Sov. J. Opt. Technol. (1)

T. M. Leushina, “Testing of Aspherical Surfaces on a Goniometer,” Sov. J. Opt. Technol. 45, 544 (1978).

Other (6)

Ref. 7, Section 3: “Aspherometer,” in Generation of Optical Surfaces, K. J. Kumanin, Ed. [The Focal Library, London, 1962 (Russian edition)], pp. 312–325.

R. S. Longhurst, Geometrical and Physical Optics (Wiley, New York, 1964), p. 68.

A. Cornejo-Rodriquez et al., “Profile Measurement of a Conic Surface Using a Nodal Bench and a He-Ne Laser,” in Conference Digest, Optics in Modern Science Technology, ICO-13, Sapporo, Japan (1984), p. 402.

W. T. Welford, R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy (Academic, New York, 1978).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972). p. 91.

B. A. Chunin, S. Kachkin, “Methods of Controlling the Shape of Aspheric Surfaces,” Sec. 1: “Mechanical Methods,” in Generation of Optical Surfaces, K. J. Kumanin, Ed. [Focal Library, London, 1962 (Russian edition)], Chap. 9, pp. 294–300.

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Figures (6)

Fig. 1
Fig. 1

Variables z, s, r, x, and θ for a conic surface are shown; pcc is the paraxial center of curvature.

Fig. 2
Fig. 2

Value of the prc is calculated by the difference between positions (a) and (b). (a) To find the vertex, the surface is rotated until the reflection pattern does not change. (b) To find the prc, the surface is rotated by a small angle until the reflected beam remains stationary.

Fig. 3
Fig. 3

Rotation and longitudinal displacement to obtain normal nonparaxial incidence on the surface. (a) When the surface is rotated by an angle θ, the reflected beam is off-axis; this means that incident beam is not normal. (b) For the same angle θ as in (a), the surface is displaced along the surface’s optical axis until the reflected beam is on-axis again.

Fig. 4
Fig. 4

(a) Diagram of the nodal bench used for measurements of the profile. (b) Actual experimental setup showing the He–Ne laser L; observing screen OS; nodal bench NB; surface under measurement S; and hair support that produces a reference diffraction pattern.

Fig. 5
Fig. 5

Typical diffraction patterns produced by the surface vertex VP and by the hair placed on the laser beam path HP and HP′. The first one is used to find the vertex position. The reflected pattern HP, coming from the front surface of the lens, under measurement, is used to center such reflection. The pattern HP′ comes from the back surface of the lens.

Fig. 6
Fig. 6

Experimental results of z and Δz vs s for the conic surface of the Cinephor lens (Bausch & Lomb).

Equations (8)

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z = c s 2 1 + [ 1 ( k + 1 ) c 2 s 2 ] 1 / 2 ,
X = k z .
tan θ = s r z ( k + 1 ) ,
R = ( 1 k c 2 s 2 ) 3 / 2 c ;
δ z = 1 | k | δ X ,
s = r tan θ .
δ s = | s r | δ r + | s θ | δ θ = tan θ δ r + r sec 2 θ δ θ ;
δ s = δ r + 2 r δ θ .

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