Abstract

Surface irregularity errors are conventionally used to specify fabrication accuracy of spherical, aspheric, or plane surfaces. However, in some cases, the amplitude of the irregularities fails to fully describe the surface accuracy requirement when the pupil size is small compared to the surface diameter. In such cases, the irregularity slope will induce distortion. A spatially dependent representation of the irregularity slope is proposed and implemented to specify the surface accuracy. As an optical design example, we study in detail the case of the front surfaces of a fish-eye lens and a panomorph lens. Panoramic lenses are characterized by a small entrance pupil and by important distortion. For both lenses, we found that the novel field-dependent mathematical descriptor provided a nearly perfect agreement with Monte Carlo analyses and can be used to specify the spatially dependent irregularity requirement. The approach is not limited to wide-angle lenses.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. S. Chahl and M. V. Srinivasan, “Range estimation with a panoramic visual sensor,” J. Opt. Soc. Am. A 14, 2144–2151(1997).
    [CrossRef]
  2. G. Jang, S. Kim, and I. Kweon, “Single-camera panoramic stereo system with single-viewpoint optics,” Opt. Lett. 31, 41–43 (2006).
    [CrossRef] [PubMed]
  3. T. J. Herbert, “Calibration of fisheye lenses by inversion of area projections,” Appl. Opt. 25, 1875–1876 (1986).
    [CrossRef] [PubMed]
  4. M. C. Sanson, C. T. Tienvieri, and S. VanKerkhove, “Localized slope errors and their impact on image performance requirements,” in International Optical Design, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper ThB2.
  5. S. Thibault, “IR panomorph lens imager and applications,” Proc. SPIE 6940, 69401A (2008).
    [CrossRef]
  6. G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
    [CrossRef]
  7. K.-L. Huang, “The tolerance analysis of wide-angle lens,” Proc. SPIE 5638, 976–985 (2005).
    [CrossRef]
  8. J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE 7433, 74330D (2009).
    [CrossRef]
  9. G. Erdei, G. Szarvas, and E. Lörincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718–728 (2004).
    [CrossRef]
  10. J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
    [CrossRef]
  11. R. H. Wilson, R. C. Brost, D. R. Strip, R. J. Sudol, R. N. Youngworth, and P. O. McLaughlin, “Considerations for tolerancing aspheric optical components,” Appl. Opt. 43, 57–66 (2004).
    [CrossRef] [PubMed]
  12. W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

2009 (1)

J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE 7433, 74330D (2009).
[CrossRef]

2008 (1)

S. Thibault, “IR panomorph lens imager and applications,” Proc. SPIE 6940, 69401A (2008).
[CrossRef]

2007 (1)

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

2006 (2)

M. C. Sanson, C. T. Tienvieri, and S. VanKerkhove, “Localized slope errors and their impact on image performance requirements,” in International Optical Design, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper ThB2.

G. Jang, S. Kim, and I. Kweon, “Single-camera panoramic stereo system with single-viewpoint optics,” Opt. Lett. 31, 41–43 (2006).
[CrossRef] [PubMed]

2005 (1)

K.-L. Huang, “The tolerance analysis of wide-angle lens,” Proc. SPIE 5638, 976–985 (2005).
[CrossRef]

2004 (2)

G. Erdei, G. Szarvas, and E. Lörincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718–728 (2004).
[CrossRef]

R. H. Wilson, R. C. Brost, D. R. Strip, R. J. Sudol, R. N. Youngworth, and P. O. McLaughlin, “Considerations for tolerancing aspheric optical components,” Appl. Opt. 43, 57–66 (2004).
[CrossRef] [PubMed]

2000 (1)

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

1999 (1)

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

1997 (1)

1986 (1)

Aikens, D. M.

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Brost, R. C.

Chahl, J. S.

English, R. E.

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Erdei, G.

G. Erdei, G. Szarvas, and E. Lörincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718–728 (2004).
[CrossRef]

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

Fodor, J.

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

Herbert, T. J.

House, W.

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Huang, K.-L.

K.-L. Huang, “The tolerance analysis of wide-angle lens,” Proc. SPIE 5638, 976–985 (2005).
[CrossRef]

Jang, G.

Kallo, P.

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

Kim, S.

Kweon, I.

Lawson, J. K.

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Lörincz, E.

G. Erdei, G. Szarvas, and E. Lörincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718–728 (2004).
[CrossRef]

McLaughlin, P. O.

Nichols, M. A.

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Parent, J.

J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE 7433, 74330D (2009).
[CrossRef]

Sanson, M. C.

M. C. Sanson, C. T. Tienvieri, and S. VanKerkhove, “Localized slope errors and their impact on image performance requirements,” in International Optical Design, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper ThB2.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

Srinivasan, M. V.

Strip, D. R.

Sudol, R. J.

Szarvas, G.

G. Erdei, G. Szarvas, and E. Lörincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718–728 (2004).
[CrossRef]

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

Thibault, S.

J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE 7433, 74330D (2009).
[CrossRef]

S. Thibault, “IR panomorph lens imager and applications,” Proc. SPIE 6940, 69401A (2008).
[CrossRef]

Tienvieri, C. T.

M. C. Sanson, C. T. Tienvieri, and S. VanKerkhove, “Localized slope errors and their impact on image performance requirements,” in International Optical Design, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper ThB2.

Ujhelyi, F.

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

VanKerkhove, S.

M. C. Sanson, C. T. Tienvieri, and S. VanKerkhove, “Localized slope errors and their impact on image performance requirements,” in International Optical Design, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper ThB2.

Whistler, W. T.

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Wilson, R. H.

Youngworth, R. N.

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Proc. SPIE (6)

S. Thibault, “IR panomorph lens imager and applications,” Proc. SPIE 6940, 69401A (2008).
[CrossRef]

G. Erdei, J. Fodor, P. Kallo, G. Szarvas, and F. Ujhelyi, “Design of high-numerical-aperture Fourier objectives for holographic memory card writing/reading equipment,” Proc. SPIE 4093, 464–473 (2000).
[CrossRef]

K.-L. Huang, “The tolerance analysis of wide-angle lens,” Proc. SPIE 5638, 976–985 (2005).
[CrossRef]

J. Parent and S. Thibault, “Tolerancing panoramic lenses,” Proc. SPIE 7433, 74330D (2009).
[CrossRef]

G. Erdei, G. Szarvas, and E. Lörincz, “Tolerancing surface accuracy of aspheric lenses used for imaging purposes,” Proc. SPIE 5249, 718–728 (2004).
[CrossRef]

J. K. Lawson, D. M. Aikens, R. E. English, W. T. Whistler, W. House, and M. A. Nichols, “Surface figure and roughness tolerances for NIF optics and the interpretation of the gradient, P-V wavefront, and RMS specifications,” Proc. SPIE 3782, 510–517 (1999).
[CrossRef]

Other (2)

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

M. C. Sanson, C. T. Tienvieri, and S. VanKerkhove, “Localized slope errors and their impact on image performance requirements,” in International Optical Design, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper ThB2.

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 (9)

Fig. 1
Fig. 1

In this layout of a fish-eye lens (FEL), the small entrance pupil diameter compared to the front lens diameter is visible.

Fig. 2
Fig. 2

In this layout of a panomorph lens (PML), the very small entrance pupil diameter compared to the front lens diameter is even more critical than in the FEL.

Fig. 3
Fig. 3

Image displacement Δ H is not constant when a Gaussian-shaped localized error is placed at different positions on the front surface of the PML. Each curve of a different color represents a different position of this error. The effect appears to be worse between 30 ° and 70 ° because the produced displacements are larger. On each curve, the maximum and minimum are located at the position of the largest positive and negative slopes in the Gaussian errors.

Fig. 4
Fig. 4

Monte-Carlo-generated error surfaces on the front surface of the PML. Each different curve represents a different Monte Carlo trial. It appears that the maximum change in resolution occurs between 30 ° and 70 ° because the changes are larger in that region.

Fig. 5
Fig. 5

Schematic of (a) before and (b) after a surface error approximated by a prism of vertex angle α modifies the object angle θ and the image height H.

Fig. 6
Fig. 6

Local focal length f ( θ ) H / θ . (a) For the FEL, the resolution decreases with the field angle. (b) For the PML, the resolution increases with the field angle, up to a maximum of around 63 ° before falling back down.

Fig. 7
Fig. 7

Resulting envelope curves showing the displacement of the image caused by a constant slope error. This displacement depends on the field angle in both lenses. (a) For the FEL, moving the error toward the edge of the front lens worsens the displacement because the effects of increasing incidence angle more than counterbalance the decreasing local focal length. (b) For the PML, because the effects of increasing incidence angle are less important than changes in local focal length, the produced displacement is shaped in great part by the local focal length or, in other words, the impacts are worse where the resolution is higher.

Fig. 8
Fig. 8

Maximum allowed surface error slopes versus the field angle to achieve a constant image height displacement of 10 μm . (a) For the FEL, the edge is the most critical region. (b) For the PML, the most critical region is near the maximum of resolution of around 60 ° .

Fig. 9
Fig. 9

Maximum allowed surface error slopes versus the lens coordinate Y to achieve a constant image height displacement of 10 μm . Both curves are similar to Fig. 8, the difference coming from the nonlinear relation between Y and θ. (a) For the FEL and (b) for the PML.

Equations (7)

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

α = arctan ( Z ideal / Y ) arctan ( Z ideal / Y + Z error / Y ) .
α [ Z error / Y ] / [ 1 + tan 2 ( φ N ) ] .
Δ θ = ( θ φ N ) α + arcsin [ ( n 2 sin 2 ( θ φ N ) ) 1 / 2 sin ( α ) cos ( α ) sin ( θ φ N ) ] .
Δ θ α ( n 1 ) [ 1 + ( θ φ N ) 2 ( n + 1 ) / ( 2 n ) + ( θ φ N ) 4 ( n + 1 ) ( 5 n 2 + 3 ) / ( 24 n 3 ) + ] .
H = a + b θ + c θ 2 + d θ 3 + O ( θ 4 ) .
Δ H = f ( θ ) Δ θ .
Δ H f ( θ ) / [ 1 + tan 2 ( φ N ) ] × ( n 1 ) [ 1 + ( θ φ N ) 2 ( n + 1 ) / ( 2 n ) + ] × Z error / Y .

Metrics