Abstract

We present a new method for precise ray-tracing simulation considering form errors in the fabrication process of aspheric lenses. The Nagata patch, a quadratic interpolant for surface meshes using normal vectors, is adopted for representing the lens geometry with mid-spectral frequencies of surface profile errors. Several improvements in the ray–patch intersection calculation and its acceleration technique are also proposed. The developed algorithm is applied to ray-tracing simulation of optical disk pick-up aspheric objectives, and this technique requires 105 to 109 times fewer patches than a polygonal approximation. The simulation takes only several seconds on a standard PC.

© 2010 Optical Society of America

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References

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  1. A. K. Rigler and T. P. Vogl, “Spline functions: an alternative representation of aspheric surfaces,” Appl. Opt. 10, 1648–1651 (1971).
    [Crossref] [PubMed]
  2. J. L. Rayces and X. Cheng, “Numerical integration of an aspheric surface profile,” Proc. SPIE 6342, 634224 (2007).
    [Crossref]
  3. T. P. Vogl, A. K. Rigler, and B. R. Canty, “Asymmetric lens design using bicubic splines: application to the color TV lighthouse,” Appl. Opt. 10, 2513–2516 (1971).
    [Crossref] [PubMed]
  4. J. E. Stacy, “Asymmetric spline surfaces: characteristics and applications,” Appl. Opt. 23, 2710–2714 (1984).
    [Crossref] [PubMed]
  5. G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical software,” Proc. SPIE 4769, 75–83(2002).
    [Crossref]
  6. A. Glassner, An Introduction to Ray Tracing (Academic, 1989).
  7. T. Nishita, T. W. Sederberg, and M. Kakimoto, “Ray tracing trimmed rational surface patches,” Comput. Graph. 24, 337–345 (1990).
    [Crossref]
  8. S. H. M. Roth, P. Diezi, and M. H. Gross, “Ray tracing triangular Bézier patches,” Comput. Graph. Forum 20, 422–430(2001).
    [Crossref]
  9. H. Chase, “Optical design with rotationally symmetric NURBS,” Proc. SPIE 4832, 10–24 (2002).
    [Crossref]
  10. T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347(2005).
    [Crossref]
  11. T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
    [Crossref]
  12. T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
    [Crossref]
  13. E. W. Weisstein, “Quartic equation,” from Wolfram MathWorld, http://mathworld.wolfram.com/QuarticEquation.html.
  14. M. S. Petković, C. Carstensen, and M. Trajkovíc, “Weierstrass formula and zero-finding methods,” Numer. Math. 69, 353–372 (1995).
    [Crossref]

2008 (2)

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

2007 (1)

J. L. Rayces and X. Cheng, “Numerical integration of an aspheric surface profile,” Proc. SPIE 6342, 634224 (2007).
[Crossref]

2005 (1)

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347(2005).
[Crossref]

2002 (2)

G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical software,” Proc. SPIE 4769, 75–83(2002).
[Crossref]

H. Chase, “Optical design with rotationally symmetric NURBS,” Proc. SPIE 4832, 10–24 (2002).
[Crossref]

2001 (1)

S. H. M. Roth, P. Diezi, and M. H. Gross, “Ray tracing triangular Bézier patches,” Comput. Graph. Forum 20, 422–430(2001).
[Crossref]

1995 (1)

M. S. Petković, C. Carstensen, and M. Trajkovíc, “Weierstrass formula and zero-finding methods,” Numer. Math. 69, 353–372 (1995).
[Crossref]

1990 (1)

T. Nishita, T. W. Sederberg, and M. Kakimoto, “Ray tracing trimmed rational surface patches,” Comput. Graph. 24, 337–345 (1990).
[Crossref]

1984 (1)

1971 (2)

Canty, B. R.

Carstensen, C.

M. S. Petković, C. Carstensen, and M. Trajkovíc, “Weierstrass formula and zero-finding methods,” Numer. Math. 69, 353–372 (1995).
[Crossref]

Chase, H.

H. Chase, “Optical design with rotationally symmetric NURBS,” Proc. SPIE 4832, 10–24 (2002).
[Crossref]

Cheng, X.

J. L. Rayces and X. Cheng, “Numerical integration of an aspheric surface profile,” Proc. SPIE 6342, 634224 (2007).
[Crossref]

Diezi, P.

S. H. M. Roth, P. Diezi, and M. H. Gross, “Ray tracing triangular Bézier patches,” Comput. Graph. Forum 20, 422–430(2001).
[Crossref]

Freniere, E. R.

G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical software,” Proc. SPIE 4769, 75–83(2002).
[Crossref]

Gardner, L. R.

G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical software,” Proc. SPIE 4769, 75–83(2002).
[Crossref]

Glassner, A.

A. Glassner, An Introduction to Ray Tracing (Academic, 1989).

Gregory, G. G.

G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical software,” Proc. SPIE 4769, 75–83(2002).
[Crossref]

Gross, M. H.

S. H. M. Roth, P. Diezi, and M. H. Gross, “Ray tracing triangular Bézier patches,” Comput. Graph. Forum 20, 422–430(2001).
[Crossref]

Hama, T.

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

Kakimoto,

T. Nishita, T. W. Sederberg, and M. Kakimoto, “Ray tracing trimmed rational surface patches,” Comput. Graph. 24, 337–345 (1990).
[Crossref]

Makinouchi, A.

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

Nagata, T.

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347(2005).
[Crossref]

Nishita, T.

T. Nishita, T. W. Sederberg, and M. Kakimoto, “Ray tracing trimmed rational surface patches,” Comput. Graph. 24, 337–345 (1990).
[Crossref]

Petkovic, M. S.

M. S. Petković, C. Carstensen, and M. Trajkovíc, “Weierstrass formula and zero-finding methods,” Numer. Math. 69, 353–372 (1995).
[Crossref]

Rayces, J. L.

J. L. Rayces and X. Cheng, “Numerical integration of an aspheric surface profile,” Proc. SPIE 6342, 634224 (2007).
[Crossref]

Rigler, A. K.

Roth, S. H. M.

S. H. M. Roth, P. Diezi, and M. H. Gross, “Ray tracing triangular Bézier patches,” Comput. Graph. Forum 20, 422–430(2001).
[Crossref]

Sederberg, T. W.

T. Nishita, T. W. Sederberg, and M. Kakimoto, “Ray tracing trimmed rational surface patches,” Comput. Graph. 24, 337–345 (1990).
[Crossref]

Stacy, J. E.

Takamura, M.

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

Takuda, H.

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

Teodosiu, C.

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

Trajkovíc, M.

M. S. Petković, C. Carstensen, and M. Trajkovíc, “Weierstrass formula and zero-finding methods,” Numer. Math. 69, 353–372 (1995).
[Crossref]

Vogl, T. P.

Weisstein, E. W.

E. W. Weisstein, “Quartic equation,” from Wolfram MathWorld, http://mathworld.wolfram.com/QuarticEquation.html.

Appl. Opt. (3)

Comput. Aided Geom. Des. (1)

T. Nagata, “Simple local interpolation of surfaces using normal vectors,” Comput. Aided Geom. Des. 22, 327–347(2005).
[Crossref]

Comput. Graph. (1)

T. Nishita, T. W. Sederberg, and M. Kakimoto, “Ray tracing trimmed rational surface patches,” Comput. Graph. 24, 337–345 (1990).
[Crossref]

Comput. Graph. Forum (1)

S. H. M. Roth, P. Diezi, and M. H. Gross, “Ray tracing triangular Bézier patches,” Comput. Graph. Forum 20, 422–430(2001).
[Crossref]

Int. J. Mech. Sci. (1)

T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, and H. Takuda, “Finite-element simulation of spring back in sheet metal forming using local interpolation for tool surfaces,” Int. J. Mech. Sci. 50, 175–192 (2008).
[Crossref]

J. Comput. Sci. Technol. (1)

T. Hama, M. Takamura, A. Makinouchi, C. Teodosiu, and H. Takuda, “Formulation of contact problems in sheet metal forming simulation using local interpolation for tool surfaces,” J. Comput. Sci. Technol. 2, 68–80 (2008).
[Crossref]

Numer. Math. (1)

M. S. Petković, C. Carstensen, and M. Trajkovíc, “Weierstrass formula and zero-finding methods,” Numer. Math. 69, 353–372 (1995).
[Crossref]

Proc. SPIE (3)

H. Chase, “Optical design with rotationally symmetric NURBS,” Proc. SPIE 4832, 10–24 (2002).
[Crossref]

J. L. Rayces and X. Cheng, “Numerical integration of an aspheric surface profile,” Proc. SPIE 6342, 634224 (2007).
[Crossref]

G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical software,” Proc. SPIE 4769, 75–83(2002).
[Crossref]

Other (2)

A. Glassner, An Introduction to Ray Tracing (Academic, 1989).

E. W. Weisstein, “Quartic equation,” from Wolfram MathWorld, http://mathworld.wolfram.com/QuarticEquation.html.

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Figures (12)

Fig. 1
Fig. 1

Nagata patch interpolation of an edge and a surface. The curvature parameter vector c is uniquely determined so that the interpolants are orthogonal to the normal vectors at the vertices.

Fig. 2
Fig. 2

Error distributions on the aspheric lens surface using the Nagata patch interpolation.

Fig. 3
Fig. 3

Convergence of errors using the linear triangle and the Nagata patch expressions.

Fig. 4
Fig. 4

Intersection of a line segment with a Nagata patch. The object envelope can be used to determine the start x a and the end x b of the line segment for testing intersections. A local coordinate system with axes e 1 , e 2 , and e 3 is constructed to find the parameters η, ζ, and λ that correspond to the intersection in the global coordinate system x y z .

Fig. 5
Fig. 5

(a) Space cells remember Nagata patches whose bounding boxes intersect the cells. (b) Cells intersecting the ray are also identified. (c) Only the cells common to (a) and (b) are searched for line–patch intersections.

Fig. 6
Fig. 6

Candidate points for identifying the bounding box of a Nagata patch. In this example, the minimum x and y coordinates are the extrema on the edges l 2 and l 1 , respectively. The minimum x and maximum y coordinates are x 00 and y 11 , respectively, both of which are the components of the vertex position vectors. An extremum on the edge l 3 in the y direction exists, but it is not taken into consideration since it is beyond the valid param eter range.

Fig. 7
Fig. 7

Comparison of computational time for the intersection analysis. Open symbols represent the time for 20,000 patches with N number of rays using the (A) double-bucketing and (B) brute-force algorithms. Solid symbols indicate the time for 10,000 rays with N number of patches using the (a) double-bucketing and (b) brute-force algorithms.

Fig. 8
Fig. 8

Refraction of a ray changes the current ray direction vector v to v . The normal vector n depends on the surface parameters η and ζ of the Nagata patch.

Fig. 9
Fig. 9

Plano–convex hyperboloid lens system, which converges incident parallel rays to a point on the optical axis. The lens consists of a planar surface s 1 and an ideal hyperboloid s 2 without spherical aberration. The lens diameter is 4.2 mm , and aspheric polynomial parameters for the surface s 2 are R = 1.838 992 mm , k = 2.550 649 647 705 , and all a i are zero. The refractive index of the lens is 1.597.

Fig. 10
Fig. 10

Compact disk pick-up head system. The system consists of an objective lens and a compact disk whose refractive indices are 1.597 and 1.573, respectively. The objective lens consists of aspheric surfaces s 3 and s 4 . Aspheric polynomial parameters for surface s 3 are R = 3.154 57 mm , k = 0.865 45 , a 4 = 1.152 × 10 3 , a 6 = 8.037 × 10 5 , a 8 = 1.795 × 10 5 , and a 10 = 7.23 × 10 7 . Parameters for surface s 4 are R = 10.235 41 mm , k = 8.798 91 , a 4 = 3.736 × 10 3 , a 6 = 3.982 × 10 4 , a 8 = 2.071 × 10 4 , and a 10 = 5.14 × 10 5 .

Fig. 11
Fig. 11

Incident ray distribution on the focal plane and the spot radii obtained from ray-tracing simulation of the plano–convex hyperboloid lens system for N number of patches. Their spot radii values for cases (c) and (d) are below the Airy radius ( 0.7156 μm ) of the optical system.

Fig. 12
Fig. 12

Comparison of the spot size convergence based on the linear polygonal and quadratic Nagata patch representations of lens surfaces.

Equations (35)

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x ( ξ ) = x 0 + ( d c ) ξ + c ξ 2 ,
c ( d , n 0 , n 1 ) = { [ n 0 , n 1 ] 1 c 2 [ 1 c c 1 ] { n 0 · d n 1 · d } ( c ± 1 ) 0 ( c = ± 1 ) ,
x ( η , ζ ) = c 00 + c 10 η + c 01 ζ + c 11 η ζ + c 20 η 2 + c 02 ζ 2 ,
c 00 = x 00 , c 10 = x 10 x 00 , c 01 = x 11 x 10 + c 1 c 3 , c 11 = c 3 + c 1 c 2 , c 20 = c 1 , c 02 = c 2 ,
z ( r ) = r 2 / R 1 + 1 ( k + 1 ) r 2 / R 2 + i = 2 a i r i ,
x ( λ ) = x a + λ d ,
x a + λ d = c 00 + c 10 η + c 01 ζ + c 11 η ζ + c 20 η 2 + c 02 ζ 2 ,
e 3 = d / d ,
e 1 = e 3 × c 02 | e 3 × c 02 | .
e 2 = e 3 × e 1 .
0 = α 00 + α 10 η + α 01 ζ + α 11 η ζ + α 20 η 2 ,
0 = β 00 + β 10 η + β 01 ζ + β 11 η ζ + β 20 η 2 + β 02 ζ 2 ,
λ d = γ 00 + γ 10 η + γ 01 ζ + γ 11 η ζ + γ 20 η 2 + γ 02 ζ 2 ,
α i j = { e 1 · ( c i j x a ) i = j = 0 e 1 · c i j otherwise ,
β i j = { e 2 · ( c i j x a ) i = j = 0 e 2 · c i j otherwise ,
γ i j = { e 3 · ( c i j x a ) i = j = 0 e 3 · c i j otherwise .
( α 01 + α 11 η ) ζ = ( α 00 + α 10 η + α 20 η 2 ) .
M η 4 + N η 3 + O η 2 + P η + Q = 0 ,
M = α 11 μ 1 + α 20 2 β 02 , N = α 01 μ 1 + α 11 μ 2 + 2 α 10 α 20 β 02 , O = α 01 μ 2 + α 11 μ 3 + β 02 ( α 10 2 + 2 α 00 α 20 ) , P = α 01 μ 3 + α 11 μ 4 + 2 α 00 α 10 β 02 , Q = α 01 μ 4 + α 00 2 β 02 ,
μ 1 = α 11 β 20 α 20 β 11 , μ 2 = α 11 β 10 α 10 β 11 + α 01 β 20 α 20 β 01 , μ 3 = α 01 β 10 α 10 β 01 + α 11 β 00 α 00 β 11 , μ 4 = α 01 β 00 α 00 β 01 .
M ( 1 / η ) 4 + N ( 1 / η ) 3 + O ( 1 / η ) 2 + P ( 1 / η ) + Q = 0 ,
α 20 η 2 + α 10 η + α 00 = 0 ,
β 02 ζ 2 + ( β 01 + β 11 η ) ζ + β 00 + β 10 η + β 20 η 2 = 0.
{ α i j β i j } = 1 α max { α i j β i j } ,
α max = max ( | α i j | , | β i j | , | γ i j | ) .
| a b | max ( | a | , | b | ) < ε ,
| a | < ε .
x ( ξ i ) ξ i = ( d i c i ) + 2 c i ξ i ,
ξ i ( j ) = ( d i ( j ) c i ( j ) ) 2 c i ( j ) ,
x ( η , ζ ) η = c 10 + c 11 ζ + 2 c 20 η , x ( η , ζ ) ζ = c 01 ζ + c 11 η + 2 c 02 ζ .
{ η ( j ) ζ ( j ) } = [ 2 c 20 ( j ) c 11 ( j ) c 11 ( j ) 2 c 02 ( j ) ] 1 { c 10 ( j ) c 01 ( j ) } .
n x = ceil ( l x a x p ) , n y = ceil ( l y a y p ) , n z = c e i l ( l z a z p ) ,
t x = floor ( n x ( x x min ) l x ) , t y = floor ( n y ( y y min ) l y ) , t z = floor ( n z ( z z min ) l z ) ,
n ( η , ζ ) = x ( η , ζ ) η × x ( η , ζ ) ζ | x ( η , ζ ) η × x ( η , ζ ) ζ | .
v = N r v + [ sgn ( cos θ ) 1 N r 2 { 1 cos 2 θ } N r cos θ ] n ,

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