Abstract

A local optical surface representation as a sum of basis functions is proposed and implemented. Specifically, we investigate the use of linear combination of Gaussians. The proposed approach is a local descriptor of shape and we show how such surfaces are optimized to represent rotationally non-symmetric surfaces as well as rotationally symmetric surfaces. As an optical design example, a single surface off-axis mirror with multiple fields is optimized, analyzed, and compared to existing shape descriptors. For the specific case of the single surface off-axis magnifier with a 3 mm pupil, >15mm eye relief, 24 degree diagonal full field of view, we found the linear combination of Gaussians surface to yield an 18.5% gain in the average MTF across 17 field points compared to a Zernike polynomial up to and including 10th order. The sum of local basis representation is not limited to circular apertures.

© 2008 Optical Society of America

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References

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  1. G. D. Wasserman and E. Wolf, "On the theory of aplanatic aspheric systems," Proc. Phys. Soc. B 62, 2-8 (1949).
    [CrossRef]
  2. J. M. Rodgers, "Nonstandard representations of aspheric surfaces in optical design," Ph.D. Thesis, University of Arizona (1984).
  3. D. Knapp, "Conformal Optical Design," Ph.D. Thesis, University of Arizona (2002).
  4. J. E. Stacy, "Design of unobscured reflective optical systems with general surfaces," Ph.D. Thesis, University of Arizona (1983).
  5. T. Davenport, "Creation of a uniform circular illuminance distribution using faceted reflective NURBS", Ph.D. Thesis, University of Arizona (2002).
  6. H. Chase, "Optical design with rotationally symmetric NURBS," Proc. SPIE 4832, 10-24 (2002).
    [CrossRef]
  7. S. Lerner, "Optical design using novel aspheric surfaces," Ph.D. Thesis, University of Arizona (2003).
  8. M. D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University Press, 2003).
    [CrossRef]
  9. G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express (15), 5218-5226 (2007).
    [CrossRef] [PubMed]
  10. O. Cakmakci and J. Rolland, "Head-worn displays: A review," J. Disp. Tech. 2, 199-216 (2006).
    [CrossRef]
  11. O. Cakmakci and J. Rolland, "Dual-element off-axis near-eye optical magnifier," Opt. Lett. 32, 1363-1365 (2007).
    [CrossRef] [PubMed]
  12. J. M. Rodgers and K. P. Thompson, "Benefits of freeform mirror surfaces in optical design," Proceedings of the American Society of Precision Engineering 2004 winter topical meeting on freeform optics, 31, 73-78 (2004).

2002 (1)

H. Chase, "Optical design with rotationally symmetric NURBS," Proc. SPIE 4832, 10-24 (2002).
[CrossRef]

1949 (1)

G. D. Wasserman and E. Wolf, "On the theory of aplanatic aspheric systems," Proc. Phys. Soc. B 62, 2-8 (1949).
[CrossRef]

Proc. Phys. Soc. B (1)

G. D. Wasserman and E. Wolf, "On the theory of aplanatic aspheric systems," Proc. Phys. Soc. B 62, 2-8 (1949).
[CrossRef]

Proc. SPIE (1)

H. Chase, "Optical design with rotationally symmetric NURBS," Proc. SPIE 4832, 10-24 (2002).
[CrossRef]

Other (10)

S. Lerner, "Optical design using novel aspheric surfaces," Ph.D. Thesis, University of Arizona (2003).

M. D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University Press, 2003).
[CrossRef]

G. W. Forbes, "Shape specification for axially symmetric optical surfaces," Opt. Express (15), 5218-5226 (2007).
[CrossRef] [PubMed]

O. Cakmakci and J. Rolland, "Head-worn displays: A review," J. Disp. Tech. 2, 199-216 (2006).
[CrossRef]

O. Cakmakci and J. Rolland, "Dual-element off-axis near-eye optical magnifier," Opt. Lett. 32, 1363-1365 (2007).
[CrossRef] [PubMed]

J. M. Rodgers and K. P. Thompson, "Benefits of freeform mirror surfaces in optical design," Proceedings of the American Society of Precision Engineering 2004 winter topical meeting on freeform optics, 31, 73-78 (2004).

J. M. Rodgers, "Nonstandard representations of aspheric surfaces in optical design," Ph.D. Thesis, University of Arizona (1984).

D. Knapp, "Conformal Optical Design," Ph.D. Thesis, University of Arizona (2002).

J. E. Stacy, "Design of unobscured reflective optical systems with general surfaces," Ph.D. Thesis, University of Arizona (1983).

T. Davenport, "Creation of a uniform circular illuminance distribution using faceted reflective NURBS", Ph.D. Thesis, University of Arizona (2002).

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Figures (2)

Fig. 1.
Fig. 1.

Illustration of a linear combination of Gaussian basis functions approximating a sphere. (Left) Original sphere and the weighted basis functions are shown. The 2D Gaussians are spaced uniformly with means centered on a 6×6 grid. Each 2D Gaussian has a variance of unity along the x and y dimensions. The weighted Gaussian basis functions shown underneath the sphere are found through least squares. (Center) 1D slice through the original function (black) and the approximation with a 6×6 grid (red). (Right) 1D slice through the original function (black) and the approximation with a 11×11 grid (red). We observe that the fit error reduces when the number of basis functions is increased.

Fig. 2.
Fig. 2.

(Left) Optical layout of the off-axis magnifier. (Center) MTF evaluated on-axis, 0.7 in the field, and at the maximum field. (Right) Interferogram of the surface represented with a linear combination of Gaussians

Tables (1)

Tables Icon

Table 1. Comparison of the transverse error function value and 17 average tangential and sagittal MTF values between an anamorphic asphere, x-y polynomial, Zernike polynomial, and a linear combination of Gaussians surface type.

Equations (6)

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z ( r ) = cr 2 1 + 1 ( 1 + k ) c 2 r 2 + i = 0 n a i r 2 i + 4 ,
z ( x , y ) = ϕ i ( x , y ) w i ,
ϕ ( x ; μ , ) = e 1 2 ( x μ ) T 1 ( x μ ) ,
Φ w = Z ,
z ( x , y ) = c ( x 2 + y 2 ) 1 + 1 ( 1 + k ) c 2 ( x 2 + y 2 ) + i ϕ i ( x , y ) w i ,
w = ( Φ T Φ ) 1 Φ T Z .

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