Abstract

In this work we show a method to obtain the profile of fast optical surfaces with symmetry of revolution, starting from experimental measurements of the longitudinal aberration and the angle between each normal line to the test surface and their optical axis. The method is based on a numerical and recursive integration, applied to a set of experimental measurements, the method gives a set of points on the test surface in orthogonal coordinates, from which we can determine the form of the surface, in independent way of a particular mathematical model that describes it. With its information is possible to adjust a polynomial to determine the paraxial radius curvature and the deformation coefficients. The method was applied to two surfaces and the results are compared with those obtained by a contact method, finding differences of some tenths of millimeter.

©2006 Optical Society of America

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References

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  1. D. MalacaraOptical Shop Testing, 2nd Ed., (John Wiley & Sons, Inc., E.U.A, 1992).
  2. R. Díaz-Uribe, A. Cornejo-Rodríguez, J. Pedraza-Contreras, O. Y. Cardona-Núñez, and A. Cordero-Davila, “Profile measurement of a conic surface, using a He-Ne laser and a nodal bench,” Appl. Opt.,  24, 2612–2615 (1985).
    [Crossref] [PubMed]
  3. R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE,  813, 355–356 (1987).
  4. R. Díaz Uribe, Pruebas ópticas por deflectometría láser (Optical testing by laser deflectometry), Ph. D. dissertation (Facultad de Ciencias UNAM, México1990).
  5. M. González Cardel, Determinación de los coeficientes de asfericidad de una superficie óptica rápida (Aspheric coefficients determination for a fast optical surface) M Sc. dissertation (Facultad de Ciencias, UNAM, México2003).
  6. O. Patenleeva Vladimirova and M. González Cardel, Métodos Numéricos (Ed. Instituto de Investigación en Tecnología Educativa de la Universidad Tecnológica de México, México2002).
  7. R. Díaz-Uribe and A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurement for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986).
    [Crossref] [PubMed]
  8. Melles — Griot, Product Catalog. http://shop.mellesgriot.com/products/optics/detail.asp?pf_id=01%20LAG%20017&plga=024192&mscssid=
  9. ZEMAX, Optical Design Program, User’s Guide Version 7.0, Focus Software, Incorporated, USA, March 1998.

1987 (1)

R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE,  813, 355–356 (1987).

1986 (1)

1985 (1)

Cardona-Núñez, O. Y.

Cordero-Davila, A.

Cornejo, A.

R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE,  813, 355–356 (1987).

Cornejo-Rodríguez, A.

Dìaz, R.

R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE,  813, 355–356 (1987).

Díaz Uribe, R.

R. Díaz Uribe, Pruebas ópticas por deflectometría láser (Optical testing by laser deflectometry), Ph. D. dissertation (Facultad de Ciencias UNAM, México1990).

Díaz-Uribe, R.

González Cardel, M.

O. Patenleeva Vladimirova and M. González Cardel, Métodos Numéricos (Ed. Instituto de Investigación en Tecnología Educativa de la Universidad Tecnológica de México, México2002).

M. González Cardel, Determinación de los coeficientes de asfericidad de una superficie óptica rápida (Aspheric coefficients determination for a fast optical surface) M Sc. dissertation (Facultad de Ciencias, UNAM, México2003).

Malacara, D.

D. MalacaraOptical Shop Testing, 2nd Ed., (John Wiley & Sons, Inc., E.U.A, 1992).

Pastrana, R.

R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE,  813, 355–356 (1987).

Patenleeva Vladimirova, O.

O. Patenleeva Vladimirova and M. González Cardel, Métodos Numéricos (Ed. Instituto de Investigación en Tecnología Educativa de la Universidad Tecnológica de México, México2002).

Pedraza-Contreras, J.

Appl. Opt. (2)

Proc. SPIE (1)

R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE,  813, 355–356 (1987).

Other (6)

R. Díaz Uribe, Pruebas ópticas por deflectometría láser (Optical testing by laser deflectometry), Ph. D. dissertation (Facultad de Ciencias UNAM, México1990).

M. González Cardel, Determinación de los coeficientes de asfericidad de una superficie óptica rápida (Aspheric coefficients determination for a fast optical surface) M Sc. dissertation (Facultad de Ciencias, UNAM, México2003).

O. Patenleeva Vladimirova and M. González Cardel, Métodos Numéricos (Ed. Instituto de Investigación en Tecnología Educativa de la Universidad Tecnológica de México, México2002).

D. MalacaraOptical Shop Testing, 2nd Ed., (John Wiley & Sons, Inc., E.U.A, 1992).

Melles — Griot, Product Catalog. http://shop.mellesgriot.com/products/optics/detail.asp?pf_id=01%20LAG%20017&plga=024192&mscssid=

ZEMAX, Optical Design Program, User’s Guide Version 7.0, Focus Software, Incorporated, USA, March 1998.

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Figures (13)

Fig. 1.
Fig. 1. The relationship between the parameters X and θ with the Cartesian coordinates s and z is shown.
Fig. 2.
Fig. 2. The intersection point Pio between the i-th normal line to the test surface S and the initial parabolic surface SA are shown.
Fig. 3.
Fig. 3. A first integration is using the trapezoid rule, to obtain approximation values, z2 ,…, zN .
Fig. 4.
Fig. 4. Intersections between normal lines and the straight lines built with successive points (z1, s1) and (z2, s2); (z2, s2) and (z3, s3). The new values s11, s21, s31, …sN1, are obtained. Further approximations are obtained in an iterative way.
Fig. 5.
Fig. 5. Differences between successive iterations, a) in a linear scale and b) in logarithmic scale to shown better the variation of last iterations
Fig. 6.
Fig. 6. Profile of the lens Cinephore obtained by numerical integration
Fig. 7.
Fig. 7. Profile of the surface LAG 017, obtained by the proposed numerical method
Fig. 8.
Fig. 8. Differences in θ angle associated to each point experimentally measured on the aspheric surfaces of the Cinephore and LAG 017 lenses.
Fig. 9.
Fig. 9. Differences in X for each experimentally measured point on the aspheric surfaces of the Cinephore and LAG 017 lenses.
Fig. 10.
Fig. 10. Sagitta differences between the best fit polynomial obtained for the NIM and CMM data for the Cinephore lens
Fig. A.1
Fig. A.1 .1 Initial alignment of the test surface with the laser
Fig. A.2
Fig. A.2 .2 After rotation of the test surface the laser beam is not retroreflected
Fig. A.3
Fig. A.3 .3 Linear displacement of the surface gives again the retroreflection condition.

Tables (7)

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Table 1. Data (θ±0.002°, X±0.01 mm) for the aspheric surface of the Cinephore lens.

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Table 2. Data of the lens Cinephore of Bausch & Lomb obtained by the method of the numerical integration.

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Table 3. Deformation coefficients for the Cinephore lens

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Table 4. Obtained experimental data of the lens LAG 017 (θ±0.002°, X±0.01 mm).

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Table 5. Data of the lens LAG 017 obtained by the method of the numerical integration

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Table 6. Deformation coefficients for the LAG017 lens

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Table 7. Paraxial curvature radius and coefficients of deformation for the surfaces characterized by the method of numerical Integration.

Equations (21)

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X = z + s tan θ ,
dz ds = tan θ .
z = tan θ ds ,
z = s 2 2 r ,
z = s tan θ + X .
s 2 2 r + s tan θ X = 0 ,
s = r ± r 2 + 2 Xr tan 2 θ tan θ .
z i + 1 = s 0 s i + 1 tan ( θ ) ds k = 0 i + 1 ( tan θ i + 1 + tan θ i 2 ) ( s i + 1 s i ) + z 0 ,
z = s s i k s i + 1 k s i k z i + 1 k + s s i + 1 k s i k s i + 1 k z i k .
s i + 1 k + 1 = { X i + 1 s i + 1 k z i k s i k z i + 1 k s i + 1 k s i k } { z i + 1 k z i k s i + 1 k s i k + 1 tan θ i + 1 } .
max { z i k + 1 z i k , s j k + 1 s j k } < ε i , j = 0 , 1 , , n ,
z = 1 2 ( 45.00 ) s 2 .
z = j = 0 5 D 2 j s 2 j ,
tan θ = j = 1 5 2 j D 2 j s 2 j 1
r = 1 2 D 2
z = s 2 2 ( 21.35 )
z i k z i k 1 s i k s i k 1 < 1 tan θ i .
z = R ± R 2 s 2 ,
s = tan θ tan 2 θ + 1 [ X R + R 2 + tan 2 θ ( 2 XR X 2 ) ] ,
z = ms ,
s = X tan θ m tan θ + 1 ,

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