Abstract

From a set of previously derived equations for conic surfaces, we derived another set of equations from which the conic constant k and the paraxial radius of curvature r can be obtained, if at least three values of the longitudinal aberration X and their corresponding angles θ of the normals to the surface are measured. The procedure is useful when the area around the vertex surface cannot be used. Some experimental results are presented.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Diaz-Uribe et al., “Profile Measurement of a Conic Surface, Using a He–Ne Laser and a Nodal Bench,” Appl. Opt. 24, 2612 (1985).
    [CrossRef] [PubMed]
  2. J. J. Kent, “Testing of Fast Concave Aspheric Reflecting Surface,” in Optical Shop Notebook, Vol. II (Optical Society of America, Washington, DC, 1979), pp 31–37.
  3. A. Cornejo-Rodriguez, D. Malacara-Hernàndez, “Caustic Coordinates in Platzeck-GaviolaTest of Conic Mirrors,” Appl. Opt. 17, 18 (1978).
    [CrossRef]
  4. 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]
  5. A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

1985

1983

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

1981

1978

Cordero-Davila, A.

Cornejo-Rodriguez, A.

Diaz-Uribe, R.

Ennos, A. E.

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

Kent, J. J.

J. J. Kent, “Testing of Fast Concave Aspheric Reflecting Surface,” in Optical Shop Notebook, Vol. II (Optical Society of America, Washington, DC, 1979), pp 31–37.

Malacara-Hernàndez, D.

Pedraza-Contreras, J.

Virdee, M. S.

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

Appl. Opt.

Proc. Soc. Photo-Opt. Instrum. Eng.

A. E. Ennos, M. S. Virdee, “Precision Measurement of Surface Form by Laser Autocollimation,” Proc. Soc. Photo-Opt. Instrum. Eng. 398, 252 (1983).

Other

J. J. Kent, “Testing of Fast Concave Aspheric Reflecting Surface,” in Optical Shop Notebook, Vol. II (Optical Society of America, Washington, DC, 1979), pp 31–37.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Variables s, z, r, X, θ, X′, and c′ for a conic surface; X′ and c′ are needed when r cannot be measured.

Fig. 2
Fig. 2

Experimental data for the Cinephor lens (Bausch & Lomb). When the data are along a straight line, the surface is a paraboloid.

Fig. 3
Fig. 3

Simulation of solving the system of Eqs. (10) for an ideal hyperboloid with k0 = −1.5 and r0 = 50 mm. Points AF, where the calculations were done, are located along the surface as shown. Each intersection of two curves gives a possible solution of Eq. (10); the best results come from points AE and BF.

Tables (2)

Tables Icon

Table I Angle θ and Longitudinal Aberration X Measured for a Bauch & Lomb Clnephor Lens; Measured prc = 44.93 mm ± 0.01 mm; the Value of k is Calculated Using Eqs. (5) and (6)

Tables Icon

Table II Calculated Longitudinal Aberration and Square Tangent of the Angle for a Hyperboloid with k0 = −1.5 and r0 = 50.0 mm (See Fig. 3)

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

z = c s 2 { 1 + [ 1 - ( k + 1 ) c 2 s 2 ] } 1 / 2 ,
tan θ = s r - z ( k + 1 ) ,
X = - k z .
{ tan 2 θ ( r k + X k + X ) 2 + X [ 2 k + ( k + 1 ) X ] } ( r k + X k + X ) 2 = 0.
k = - B ± B 2 - 4 A C 2 A ,
A = tan 2 θ ( r + X ) 2 , B = 2 X ( r + X ) tan 2 θ + 2 X r + X 2 , C = X 2 ( tan 2 θ + 1 ) . }
X = c - X ,
( X - c ) [ ( k + 1 ) tan 2 θ + 1 ] [ ( X - c ) ( k + 1 ) + 2 k r ] + k 2 r 2 tan 2 θ = 0 ,
X = r 2 tan 2 θ + c ;
r = ( X 2 - X 1 ) ( k + 1 k ) × { 1 1 + ( k + 1 ) tan 2 θ 1 - 1 1 + ( k + 1 ) tan 2 θ 2 } - 1 , r = ( X 3 - X 2 ) ( k + 1 k ) × { 1 1 + ( k + 1 ) tan 2 θ 2 - 1 1 + ( k + 1 ) tan 2 θ 3 } - 1 , }

Metrics