Abstract

A method for testing the profiles of spherical surfaces is presented. It consists of measuring the transversal deflection of a reflected He–Ne laser beam when the surface is rotated around an axis located near its center of curvature. A set of formulas that enables us to calculate the shape of the profile as well as the decentering of the rotation axis is obtained. By using a simple experimental setup, we found the differences between the experimental profile with respect to the ideal one; the accuracy that was obtained is ~3 μm. The method may be improved and is useful for convex as well as for concave surfaces. With minor modifications it is possible to test large surfaces and weak aspherics.

© 1993 Optical Society of America

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References

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  1. J. D. Evans, “Method for approximating the radius of curvature of small concave spherical mirrors using a He–Ne laser,” Appl. Opt. 10, 995–996 (1971).
  2. J. D. Evans, “Error analysis to: Method for approximating the radius of curvature of small concave spherical mirrors using a He–Ne laser,” Appl. Opt. 11, 945–946 (1972).
    [Crossref] [PubMed]
  3. F. M. Smolka, T. P. Caudell, “Surface profile measurement and angular deflection monitoring using a scanning laser beam: a noncontact method,” Appl. Opt. 17, 3284–3289 (1978).
    [Crossref] [PubMed]
  4. A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
    [Crossref]
  5. A. E. Ennos, M. S. Virdee, “Precision measurement of surface form by laser autocollimation,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 252–257 (1983).
  6. F. V. Kowalski, T. E. Milner, M. J. Stanich, “Beam deflection as a method for testing optical components,” Appl. Opt. 25, 3735–3739 (1986).
    [Crossref] [PubMed]
  7. G. Hausler, G. Schneider, “Testing optics by experimental ray tracing with a lateral effect photodiode,” Appl. Opt. 27, 5160–5164 (1988).
    [Crossref] [PubMed]
  8. J. R. Díaz-Uribe, A. Cornejo-Rodríguez, J. Pedraza-Contreras, O. Cardona-Núnez, A. Cordero-Dávila, “Profile measurement of a conic surface, using a He–Ne laser and a nodal bench,” Appl. Opt. 24, 2612–2615 (1985).
    [Crossref] [PubMed]
  9. R. Díaz-Uribe, J. Pedraza-Contreras, O. Cardona-Núnez, A. Cordero-Dávila, A. Cornejo-Rodríguez, “Cylindrical lenses: testing and radius of curvature measurement,” Appl. Opt. 25, 1707–1709 (1986).
    [Crossref] [PubMed]
  10. R. Díaz-Uribe, A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurements for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986).
    [Crossref] [PubMed]
  11. A. Cornejo-Rodríguez, A. Cordero-Dávila, “Measurement of radii of curvature of convex and concave surfaces using a nodal bench and He–Ne laser,” Appl. Opt. 19, 1743–1745 (1980).
    [Crossref] [PubMed]
  12. M. Rosete-Aguilar, “Prueba de superficies esféricas por reflexión de un haz de láser” (“Spherical surface testing by reflection of a laser beam”), B.S. thesis (Facultad de Ciencias, Universidad Nacional Autónoma de México, Del. Coyoacán, México, 1989).
  13. M. Rosete-Aguilar, R. Díaz-Uribe, “Spherical surface testing by laser deflectometry,” in Annual Meeting, Vol. 18 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 236.
  14. K. Creath, J. Wyant, “Holographic and speckle tests,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 15, p. 612.
  15. Ref. 14, Chap. 16, p. 653.
  16. Ref. 14, Chap. 17, p. 688.

1988 (1)

1986 (3)

1985 (1)

1982 (1)

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[Crossref]

1980 (1)

1978 (1)

1972 (1)

1971 (1)

Cardona-Núnez, O.

Caudell, T. P.

Cordero-Dávila, A.

Cornejo-Rodríguez, A.

Creath, K.

K. Creath, J. Wyant, “Holographic and speckle tests,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 15, p. 612.

Díaz-Uribe, J. R.

Díaz-Uribe, R.

Ennos, A. E.

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[Crossref]

A. E. Ennos, M. S. Virdee, “Precision measurement of surface form by laser autocollimation,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 252–257 (1983).

Evans, J. D.

Hausler, G.

Kowalski, F. V.

Milner, T. E.

Pedraza-Contreras, J.

Rosete-Aguilar, M.

M. Rosete-Aguilar, “Prueba de superficies esféricas por reflexión de un haz de láser” (“Spherical surface testing by reflection of a laser beam”), B.S. thesis (Facultad de Ciencias, Universidad Nacional Autónoma de México, Del. Coyoacán, México, 1989).

M. Rosete-Aguilar, R. Díaz-Uribe, “Spherical surface testing by laser deflectometry,” in Annual Meeting, Vol. 18 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 236.

Schneider, G.

Smolka, F. M.

Stanich, M. J.

Virdee, M. S.

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[Crossref]

A. E. Ennos, M. S. Virdee, “Precision measurement of surface form by laser autocollimation,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 252–257 (1983).

Wyant, J.

K. Creath, J. Wyant, “Holographic and speckle tests,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 15, p. 612.

Appl. Opt. (9)

F. V. Kowalski, T. E. Milner, M. J. Stanich, “Beam deflection as a method for testing optical components,” Appl. Opt. 25, 3735–3739 (1986).
[Crossref] [PubMed]

G. Hausler, G. Schneider, “Testing optics by experimental ray tracing with a lateral effect photodiode,” Appl. Opt. 27, 5160–5164 (1988).
[Crossref] [PubMed]

J. R. Díaz-Uribe, A. Cornejo-Rodríguez, J. Pedraza-Contreras, O. Cardona-Núnez, A. Cordero-Dávila, “Profile measurement of a conic surface, using a He–Ne laser and a nodal bench,” Appl. Opt. 24, 2612–2615 (1985).
[Crossref] [PubMed]

R. Díaz-Uribe, J. Pedraza-Contreras, O. Cardona-Núnez, A. Cordero-Dávila, A. Cornejo-Rodríguez, “Cylindrical lenses: testing and radius of curvature measurement,” Appl. Opt. 25, 1707–1709 (1986).
[Crossref] [PubMed]

R. Díaz-Uribe, A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurements for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986).
[Crossref] [PubMed]

A. Cornejo-Rodríguez, A. Cordero-Dávila, “Measurement of radii of curvature of convex and concave surfaces using a nodal bench and He–Ne laser,” Appl. Opt. 19, 1743–1745 (1980).
[Crossref] [PubMed]

J. D. Evans, “Method for approximating the radius of curvature of small concave spherical mirrors using a He–Ne laser,” Appl. Opt. 10, 995–996 (1971).

J. D. Evans, “Error analysis to: Method for approximating the radius of curvature of small concave spherical mirrors using a He–Ne laser,” Appl. Opt. 11, 945–946 (1972).
[Crossref] [PubMed]

F. M. Smolka, T. P. Caudell, “Surface profile measurement and angular deflection monitoring using a scanning laser beam: a noncontact method,” Appl. Opt. 17, 3284–3289 (1978).
[Crossref] [PubMed]

Precis. Eng. (1)

A. E. Ennos, M. S. Virdee, “High accuracy profile measurement of quasi-conical mirror surfaces by laser autocollimation,” Precis. Eng. 4, 5–8 (1982).
[Crossref]

Other (6)

A. E. Ennos, M. S. Virdee, “Precision measurement of surface form by laser autocollimation,” in Industrial Applications of Laser Technology, W. F. Fagan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.398, 252–257 (1983).

M. Rosete-Aguilar, “Prueba de superficies esféricas por reflexión de un haz de láser” (“Spherical surface testing by reflection of a laser beam”), B.S. thesis (Facultad de Ciencias, Universidad Nacional Autónoma de México, Del. Coyoacán, México, 1989).

M. Rosete-Aguilar, R. Díaz-Uribe, “Spherical surface testing by laser deflectometry,” in Annual Meeting, Vol. 18 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 236.

K. Creath, J. Wyant, “Holographic and speckle tests,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992), Chap. 15, p. 612.

Ref. 14, Chap. 16, p. 653.

Ref. 14, Chap. 17, p. 688.

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Figures (11)

Fig. 1
Fig. 1

Scanning a surface with a laser beam by rotating the surface around the z axis; the set of points scanned belongs to a curve called the profile. The incident laser beam is along the x axis.

Fig. 2
Fig. 2

Diagram for obtaining the surface profile shape equation in polar coordinates: I, incident laser beam; R, reflected beam; N, normal to the surface at the point of incidence P; ϕ, deflection angle; i, the usual angle of incidence; r, θ, polar coordinates of P. It must be clear that r is not necessarily the radius of curvature.

Fig. 3
Fig. 3

Schematic view of the experimental setup: S, test surface; BS, pellicle beam splitter; L, positive collecting lens; AP, aluminum-coated roof prism used in external reflection and located at the focal plane of lens L; PM1, PM2, and PM3, plane mirrors, the first of which produces the reference (direction) beam, while the other two send the split beam at AP to detector D; R, R′, reflected beams before and after the lens, respectively.

Fig. 4
Fig. 4

Picture of the actual setup that was used in the experiments: L, laser head; BS1 and BS2, beam splitters, the first of which is used to separate a beam for another experiment; PM1, reference plane mirror; S, mount on which the surface is located; L, collecting lens; AP, roof Amici prism that was used in external reflection; PM2 and PM3, two plane mirrors that were used to send the two divided beams to detector D; B, one of the two blocking screens; A, detector amplifier and analog display.

Fig. 5
Fig. 5

Schematic diagram for deducing the equation that relates the transversal deflection y to deflection angle ϕ: S, test surface; BS, beam splitter; L, collecting lens. In the most general case, the measuring plane is not at the focal plane. R is the beam reflected on the surface and going to the lens, R′ is the same beam behind the lens, and RB is the reference beam.

Fig. 6
Fig. 6

Plot of the measured transversal deflection y. These are averaged data after six single measurements. The error bars are the standard deviation of the mean of each set of data; see Table 1.

Fig. 7
Fig. 7

Plot of the computed scaled radius e of the test surface that was obtained by using Eq. (10). The error bars are the propagated uncertainties according to Eqs. (12) and (13). The variations from the unit value are due mainly to decentration of the surface.

Fig. 8
Fig. 8

Polar radius variation compared with the ideal surface.

Fig. 9
Fig. 9

Transversal deflection for the same profile but only single measurements. For plots 1, and 3 the scan was made in the forward direction; for plots 2 and 4 the scan was reversed. For plot 1 the decentration is similar to that in Figs. 68; for the other plots the decentration was diminished. For plots 3 and 4 the surface was rotated 180° around its optical axis (x axis).

Fig. 10
Fig. 10

Scaled polar radius that was calculated by using the data shown in Fig. 8. Note that the decentration correction gives less variation of the scaled radius, as expected.

Fig. 11
Fig. 11

Differences between the real and ideal profiles. As can be seen the different experiments show a reproducibility of the measurements to within 0.002 mm.

Tables (1)

Tables Icon

Table 1 Experimental Data for the Tested Concave Spherical Surfacea

Equations (20)

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y ( x ) = x 0 x tan [ ϕ ( x ) 2 ] d x ,
i = ϕ / 2 ,
tan i = P B PB = 1 r d r d θ .
r ( θ 2 ) = r ( θ 1 ) exp { θ 1 θ 2 tan [ ϕ ( θ ) / 2 ] d θ } .
tan ϕ = y f l ( l f ) l f .
tan ϕ = y f .
( x a ) 2 + ( y b ) 2 = r 0 2 ,
r ( θ , a , b , r 0 ) = a cos θ + b sin θ ± [ r 0 2 ( a sin θ b cos θ ) 2 ] 1 / 2 .
a = 0 . 200 ± 0 . 060 mm , b = 0 . 017 ± 0 . 009 mm ,
r ( θ ) = r ( θ 1 ) exp { i = 1 N [ tan ( ϕ i / 2 ) + tan ( ϕ i + 1 / 2 ) ] ( α / 2 ) } = r 1 e ( ϕ i , α ) ,
δ r = r 1 δ e + e δ r 1 .
δ e = i = 1 N | e ϕ i | δ ϕ i + | e α | δα = e ( θ ) [ i = 1 N | sec 2 ( ϕ i / 2 ) + sec 2 ( ϕ i + 1 / 2 ) | × αδ ϕ i 4 + i = 1 N | tan ( ϕ i / 2 ) + tan ( ϕ i + 1 / 2 ) | × δα 2 ] .
δ ϕ = | ϕ y | δ y + | ϕ f | δ f ,
ϕ y = f f 2 + y 2 ,
ϕ f = y f 2 + y 2 .
Δ r = r T r E = r 1 ( e T e E ) = r 1 Δ e ,
δ ( Δ r ) = r 1 δ ( e T e E ) + | Δ e | δ r 1 = r 1 δ e E + | Δ e | δ r 1 ,
δ ϕ δ y f .
δ e = N α 2 f δ y max + N y max 2 f δα .
δ e = 1 2 f [ Δ θδ y max + N y max δα ] .

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