Abstract

Using an optical setup that includes a square array of 3×3 holes, we used nine meridional rays to measure the effective focal length of a lens. We observed the selected meridional rays as a spot pattern on a diffuse screen. First, we generated a regular square spot pattern (reference pattern) without a lens to test, and then we generated two spot patterns in two different axial positions when the lens being tested refracts the rays. By selecting two sets of four rays of each spot pattern, we were able to measure the difference of the longitudinal (primary) spherical aberration in two positions. With this difference we were able to improve the calculation of the effective focal length. To determine the method’s precision, we first simulated the relative error in the effective focal length considering the error in the measurement of the ray heights. Then we determined the experimental relative error by means of the standard deviation of the focal lengths obtained for each spot (in the image of reference and for the images at the two different locations) for both sets of four spots. The experimental results agree very well with the simulation. The error analysis allows us to establish under what conditions it is possible to obtain relative errors of less than 1% in the effective focal length.

© 2013 Optical Society of America

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References

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  1. R. Kingslake, “A new bench for testing photographic lenses,” J. Opt. Soc. Am. 22, 207–222 (1932).
    [CrossRef]
  2. G. Smith and D. A. Atchison, “Focimeters,” in The Eye and Visual Optical Instruments (Cambridge University, 1997) Chap. 27.
  3. Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, 3rd ed. (Wiley, 2007).
  4. E. G. Ronald, “Measurement of focal length using an optical power meter,” Appl. Opt.34, Engineering Laboratory Notes, 8054–8055 (1995).
  5. D. Malacara and Z. Malacara, Handbook of Lens Design, 2nd ed. (Marcel Dekker, 2004).
  6. O. E. Olarte and Y. Mejía, “A morphological based method to calculate the centroid spots of Hartmann patterns,” Opt. Commun. 260, 87–90 (2006).
    [CrossRef]
  7. N. Zon, O. Srour, and E. N. Ribak, “Hartmann-Shack analysis errors,” Opt. Express 14, 635–643 (2006).
    [CrossRef]
  8. X. Ma, C. Rao, and H. Zheng, “Error analysis of CCD-based point source centroid computation under the background light,” Opt. Express 17, 8525–8541 (2009).
    [CrossRef]
  9. A. Patwardhan, “Subpixel position measurement using 1D, 2D, and 3D centroid algorithms with emphasis on applications in confocal microscopy,” J. Microsc. 186, 246–257 (1997).
    [CrossRef]
  10. F. A. Jenkins and H. E. White, “Plane surfaces and prism,” in Fundamentals of Optics, 4th ed. (McGraw-Hill, 1957) Chap. 2.
  11. A. E. Conrady, “Spherical aberration,” in Applied Optics and Optical Design, Part I (Dover, 1985) Chap. 2.
  12. D. Malacara and Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with only four holes,” Opt. Eng. 31, 1551–1555 (1992).
    [CrossRef]

2009 (1)

2006 (2)

O. E. Olarte and Y. Mejía, “A morphological based method to calculate the centroid spots of Hartmann patterns,” Opt. Commun. 260, 87–90 (2006).
[CrossRef]

N. Zon, O. Srour, and E. N. Ribak, “Hartmann-Shack analysis errors,” Opt. Express 14, 635–643 (2006).
[CrossRef]

1997 (1)

A. Patwardhan, “Subpixel position measurement using 1D, 2D, and 3D centroid algorithms with emphasis on applications in confocal microscopy,” J. Microsc. 186, 246–257 (1997).
[CrossRef]

1992 (1)

D. Malacara and Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with only four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

1932 (1)

Atchison, D. A.

G. Smith and D. A. Atchison, “Focimeters,” in The Eye and Visual Optical Instruments (Cambridge University, 1997) Chap. 27.

Conrady, A. E.

A. E. Conrady, “Spherical aberration,” in Applied Optics and Optical Design, Part I (Dover, 1985) Chap. 2.

Jenkins, F. A.

F. A. Jenkins and H. E. White, “Plane surfaces and prism,” in Fundamentals of Optics, 4th ed. (McGraw-Hill, 1957) Chap. 2.

Kingslake, R.

Ma, X.

Malacara, D.

D. Malacara and Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with only four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

D. Malacara and Z. Malacara, Handbook of Lens Design, 2nd ed. (Marcel Dekker, 2004).

Malacara, Z.

D. Malacara and Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with only four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

D. Malacara and Z. Malacara, Handbook of Lens Design, 2nd ed. (Marcel Dekker, 2004).

Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, 3rd ed. (Wiley, 2007).

Mejía, Y.

O. E. Olarte and Y. Mejía, “A morphological based method to calculate the centroid spots of Hartmann patterns,” Opt. Commun. 260, 87–90 (2006).
[CrossRef]

Olarte, O. E.

O. E. Olarte and Y. Mejía, “A morphological based method to calculate the centroid spots of Hartmann patterns,” Opt. Commun. 260, 87–90 (2006).
[CrossRef]

Patwardhan, A.

A. Patwardhan, “Subpixel position measurement using 1D, 2D, and 3D centroid algorithms with emphasis on applications in confocal microscopy,” J. Microsc. 186, 246–257 (1997).
[CrossRef]

Rao, C.

Ribak, E. N.

Ronald, E. G.

E. G. Ronald, “Measurement of focal length using an optical power meter,” Appl. Opt.34, Engineering Laboratory Notes, 8054–8055 (1995).

Smith, G.

G. Smith and D. A. Atchison, “Focimeters,” in The Eye and Visual Optical Instruments (Cambridge University, 1997) Chap. 27.

Srour, O.

White, H. E.

F. A. Jenkins and H. E. White, “Plane surfaces and prism,” in Fundamentals of Optics, 4th ed. (McGraw-Hill, 1957) Chap. 2.

Zheng, H.

Zon, N.

J. Microsc. (1)

A. Patwardhan, “Subpixel position measurement using 1D, 2D, and 3D centroid algorithms with emphasis on applications in confocal microscopy,” J. Microsc. 186, 246–257 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

O. E. Olarte and Y. Mejía, “A morphological based method to calculate the centroid spots of Hartmann patterns,” Opt. Commun. 260, 87–90 (2006).
[CrossRef]

Opt. Eng. (1)

D. Malacara and Z. Malacara, “Testing and centering of lenses by means of a Hartmann test with only four holes,” Opt. Eng. 31, 1551–1555 (1992).
[CrossRef]

Opt. Express (2)

Other (6)

F. A. Jenkins and H. E. White, “Plane surfaces and prism,” in Fundamentals of Optics, 4th ed. (McGraw-Hill, 1957) Chap. 2.

A. E. Conrady, “Spherical aberration,” in Applied Optics and Optical Design, Part I (Dover, 1985) Chap. 2.

G. Smith and D. A. Atchison, “Focimeters,” in The Eye and Visual Optical Instruments (Cambridge University, 1997) Chap. 27.

Z. Malacara, “Angle, prisms, curvature, and focal length measurements,” in Optical Shop Testing, 3rd ed. (Wiley, 2007).

E. G. Ronald, “Measurement of focal length using an optical power meter,” Appl. Opt.34, Engineering Laboratory Notes, 8054–8055 (1995).

D. Malacara and Z. Malacara, Handbook of Lens Design, 2nd ed. (Marcel Dekker, 2004).

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Figures (7)

Fig. 1.
Fig. 1.

Schemes for measuring the focal length in different cases: (a) positive lens, and planes z1 and z2 before the focal point, (b) positive lens, and planes z1 and z2 after the focal point, (c) positive lens, and plane z1 before focal point and plane z2 after the focal point, and (d) negative lens.

Fig. 2.
Fig. 2.

(a) Optical setup for measuring the focal length of a lens L and (b) square array of nine holes in the opaque screen HS. In DS, the refraction of the rays selected by HS is seen as a spot pattern.

Fig. 3.
Fig. 3.

Image sequence of the spot patterns for a positive lens (achromatic) of nominal focal length 200 mm. (a) Reference spot pattern (without lens under test). The spatial period of this pattern is 7.5 mm. (b) Spot pattern at location z1 (40mm from the lens), and (c)–(f) spot patterns at location z2 (50mm, 80 mm, 110 mm, and 140 mm from the lens).

Fig. 4.
Fig. 4.

Set of spots to select the meridional rays for calculating the focal length.

Fig. 5.
Fig. 5.

Spot pattern with pincushion distortion generated by a plano-convex lens of focal length 40 mm and diameter 25.4 mm, in the scheme of Fig. 1(b). The relative distortion is about +23% [5]. The size of the image window is 240×240 pixels.

Fig. 6.
Fig. 6.

Simulation of the relative error εf in the focal length measurement for a lens of focal length 200 mm when an incident ray is 7.5 mm high and 7.52mm high.

Fig. 7.
Fig. 7.

Simulation of the relative error εf in the focal length measurement for three lenses of focal length 100, 75, and 40 mm. The height of the incident ray is 4.0 mm.

Tables (9)

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Table 1. Lenses Tested

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Table 2. Focal Length Measurement as a Function of Separation Distance Δz for the Lens Number 1. Height of the Incident Ray for p is 7.5 mm and for l is 10.6 mm

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Table 3. Focal Length Measurement for Lens Number 1 When the Separation Distance is Δz=10mm

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Table 4. Focal Length Measurement for Lens Number 1 When the Separation Distance is Δz=100mm

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Table 5. Focal Length Measurement for Lens Number 2 When the Separation Distance is Δz=60mm

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Table 6. Focal Length Measurement for Lens Number 3 When the Separation Distance is Δz=110mm

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Table 7. Focal Length Measurement for Lens Number 4 When the Separation Distance is Δz=140mm

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Table 8. Focal Length Measurement for Lens Number 5 When the Separation Distance is Δz=35mm

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Table 9. Effective Focal Length for the Five Lenses Tested

Equations (6)

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

|f|=Δzπ4DΦE2ΦE1.
f=Δz(y0y1y2).
z1=f(1y1y0).
f=f(p)+Δfpl,
εf=δf|f|=δ(Δz)|Δz|+δ(y0)|y0|+δ(Δy)|Δy|.
εf=δ(y0)|y0|+1|Δz|(δ(Δz)+|f||y0|δ(Δy)).

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