Abstract

We introduce a representation of aspheric surfaces that is based on a B-spline quasi-interpolation scheme. The scheme is implemented in a ray trace algorithm, and bounds on the approximation error are established. Examples for the reproduction of aspheric surfaces in polynomial description and the ray tracing accuracy are presented. The proposed approach allows the specification of local and global structures and the efficient treatment of measured surface data. The representation gives access to a wavelet analysis, offering extended possibilities for the tolerance analysis of optical systems containing aspheric elements.

© 2011 Optical Society of America

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References

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  1. A. W. Greynolds, “Superconic and subconic surface descriptions in optical design,” Proc. SPIE 4832, 1–9 (2002).
    [CrossRef]
  2. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15, 5218–5226 (2007).
    [CrossRef] [PubMed]
  3. K. P. Thompson, F. Fournier, J. P. Rolland, and G. W. Forbes, “The Forbes polynomial: a more predictable surface for fabricators,” in Optical Fabrication and Testing (Optical Society of America, 2010), paper OTuA6.
  4. S. A. Lerner and J. M. Sasian, “Use of implicitly defined optical surfaces for the design of imaging and illumination systems,” Opt. Eng. 39, 1796–1801 (2000).
    [CrossRef]
  5. S. A. Lerner and J. M. Sasian, “Optical design with parametrically defined aspheric surfaces,” Appl. Opt. 39, 5205–5213(2000).
    [CrossRef]
  6. G. G. Gregory, E. R. Freniere, and L. R. Gardner, “Using spline surfaces in optical design software,” Proc. SPIE 4769, 75–83 (2002).
    [CrossRef]
  7. A. K. Rigler and T. P. Vogl, “Spline functions: an alternative representation of aspheric surfaces,” Appl. Opt. 10, 1648–1651 (1971).
    [CrossRef] [PubMed]
  8. J. E. Stacy, “Asymmetric spline surfaces: characteristics and applications,” Appl. Opt. 23, 2710–2714 (1984).
    [CrossRef] [PubMed]
  9. H. Chase, “Optical design with rotationally symmetric NURBS,” Proc. SPIE 4832, 10–24 (2002).
    [CrossRef]
  10. P. Ott, “Optic design of head-up displays with freeform surfaces specified by NURBS,” Proc. SPIE 7100, 71000Y (2008).
    [CrossRef]
  11. O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16, 1583–1589 (2008).
    [CrossRef] [PubMed]
  12. S. Morita, Y. Nishidate, T. Nagata, Y. Yamagata, and C. Teodosiu, “Ray-tracing simulation method using piecewise quadratic interpolant for aspheric optical systems,” Appl. Opt. 49, 3442–3451 (2010).
    [CrossRef] [PubMed]
  13. C. de Boor, A Practical Guide to Splines, revised ed., Vol.  27 of Applied Mathematical Sciences (Springer-Verlag, 2001).
  14. A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
    [CrossRef]
  15. S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).
  16. P. Jester, C. Menke, and K. Urban, “Wavelet methods for the representation, analysis and simulation of optical surfaces,” submitted to IMA J. Appl. Math.
  17. K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University, 2009).
  18. R. A. Adams, Sobolev Spaces, Vol.  65 of Pure and Applied Mathematics (Academic, 1975).
  19. K. Bittner and K. Urban, “Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions,” Appl. Comput. Harmon. Anal. 24, 94–119(2008).
    [CrossRef]
  20. Flexible Library for Efficient Numerical Solutions and Library for Adaptive Wavelet Applications, Institute for Numerical Mathematics, Ulm University.
  21. Optische Analyse und Synthese, Carl Zeiss AG.
  22. P. Jester, “Beschreibung optischer Grenzflächen mit Wavelets,” Diploma thesis, Institute for Numerical Mathematics (Ulm University, Germany, 2007).
  23. A. Dogariu, J. Uozumi, and T. Asakura, “Wavelet transform analysis of slightly rough surfaces,” Opt. Commun. 107, 1–5(1994).
    [CrossRef]
  24. C.-L. Tien and Y.-R. Lyu, “Optical surface flatness recognized by discrete wavelet transform and grey level co-occurrence matrix,” Meas. Sci. Technol. 17, 2299–2305 (2006).
    [CrossRef]

2010 (1)

2008 (3)

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16, 1583–1589 (2008).
[CrossRef] [PubMed]

P. Ott, “Optic design of head-up displays with freeform surfaces specified by NURBS,” Proc. SPIE 7100, 71000Y (2008).
[CrossRef]

K. Bittner and K. Urban, “Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions,” Appl. Comput. Harmon. Anal. 24, 94–119(2008).
[CrossRef]

2007 (1)

2006 (1)

C.-L. Tien and Y.-R. Lyu, “Optical surface flatness recognized by discrete wavelet transform and grey level co-occurrence matrix,” Meas. Sci. Technol. 17, 2299–2305 (2006).
[CrossRef]

2002 (3)

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

A. W. Greynolds, “Superconic and subconic surface descriptions in optical design,” Proc. SPIE 4832, 1–9 (2002).
[CrossRef]

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

2000 (2)

S. A. Lerner and J. M. Sasian, “Optical design with parametrically defined aspheric surfaces,” Appl. Opt. 39, 5205–5213(2000).
[CrossRef]

S. A. Lerner and J. M. Sasian, “Use of implicitly defined optical surfaces for the design of imaging and illumination systems,” Opt. Eng. 39, 1796–1801 (2000).
[CrossRef]

1994 (1)

A. Dogariu, J. Uozumi, and T. Asakura, “Wavelet transform analysis of slightly rough surfaces,” Opt. Commun. 107, 1–5(1994).
[CrossRef]

1992 (1)

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

1984 (1)

1971 (1)

Adams, R. A.

R. A. Adams, Sobolev Spaces, Vol.  65 of Pure and Applied Mathematics (Academic, 1975).

Asakura, T.

A. Dogariu, J. Uozumi, and T. Asakura, “Wavelet transform analysis of slightly rough surfaces,” Opt. Commun. 107, 1–5(1994).
[CrossRef]

Bittner, K.

K. Bittner and K. Urban, “Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions,” Appl. Comput. Harmon. Anal. 24, 94–119(2008).
[CrossRef]

Cakmakci, O.

Chase, H.

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

Cohen, A.

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Daubechies, I.

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

de Boor, C.

C. de Boor, A Practical Guide to Splines, revised ed., Vol.  27 of Applied Mathematical Sciences (Springer-Verlag, 2001).

Dogariu, A.

A. Dogariu, J. Uozumi, and T. Asakura, “Wavelet transform analysis of slightly rough surfaces,” Opt. Commun. 107, 1–5(1994).
[CrossRef]

Feauveau, J.-C.

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Forbes, G. W.

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

K. P. Thompson, F. Fournier, J. P. Rolland, and G. W. Forbes, “The Forbes polynomial: a more predictable surface for fabricators,” in Optical Fabrication and Testing (Optical Society of America, 2010), paper OTuA6.

Foroosh, H.

Fournier, F.

K. P. Thompson, F. Fournier, J. P. Rolland, and G. W. Forbes, “The Forbes polynomial: a more predictable surface for fabricators,” in Optical Fabrication and Testing (Optical Society of America, 2010), paper OTuA6.

Freniere, E. R.

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

Gregory, G. G.

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

Greynolds, A. W.

A. W. Greynolds, “Superconic and subconic surface descriptions in optical design,” Proc. SPIE 4832, 1–9 (2002).
[CrossRef]

Jester, P.

P. Jester, C. Menke, and K. Urban, “Wavelet methods for the representation, analysis and simulation of optical surfaces,” submitted to IMA J. Appl. Math.

P. Jester, “Beschreibung optischer Grenzflächen mit Wavelets,” Diploma thesis, Institute for Numerical Mathematics (Ulm University, Germany, 2007).

Lerner, S. A.

S. A. Lerner and J. M. Sasian, “Use of implicitly defined optical surfaces for the design of imaging and illumination systems,” Opt. Eng. 39, 1796–1801 (2000).
[CrossRef]

S. A. Lerner and J. M. Sasian, “Optical design with parametrically defined aspheric surfaces,” Appl. Opt. 39, 5205–5213(2000).
[CrossRef]

Lyu, Y.-R.

C.-L. Tien and Y.-R. Lyu, “Optical surface flatness recognized by discrete wavelet transform and grey level co-occurrence matrix,” Meas. Sci. Technol. 17, 2299–2305 (2006).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

Menke, C.

P. Jester, C. Menke, and K. Urban, “Wavelet methods for the representation, analysis and simulation of optical surfaces,” submitted to IMA J. Appl. Math.

Moore, B.

Morita, S.

Nagata, T.

Nishidate, Y.

Ott, P.

P. Ott, “Optic design of head-up displays with freeform surfaces specified by NURBS,” Proc. SPIE 7100, 71000Y (2008).
[CrossRef]

Rigler, A. K.

Rolland, J. P.

O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express 16, 1583–1589 (2008).
[CrossRef] [PubMed]

K. P. Thompson, F. Fournier, J. P. Rolland, and G. W. Forbes, “The Forbes polynomial: a more predictable surface for fabricators,” in Optical Fabrication and Testing (Optical Society of America, 2010), paper OTuA6.

Sasian, J. M.

S. A. Lerner and J. M. Sasian, “Optical design with parametrically defined aspheric surfaces,” Appl. Opt. 39, 5205–5213(2000).
[CrossRef]

S. A. Lerner and J. M. Sasian, “Use of implicitly defined optical surfaces for the design of imaging and illumination systems,” Opt. Eng. 39, 1796–1801 (2000).
[CrossRef]

Stacy, J. E.

Teodosiu, C.

Thompson, K. P.

K. P. Thompson, F. Fournier, J. P. Rolland, and G. W. Forbes, “The Forbes polynomial: a more predictable surface for fabricators,” in Optical Fabrication and Testing (Optical Society of America, 2010), paper OTuA6.

Tien, C.-L.

C.-L. Tien and Y.-R. Lyu, “Optical surface flatness recognized by discrete wavelet transform and grey level co-occurrence matrix,” Meas. Sci. Technol. 17, 2299–2305 (2006).
[CrossRef]

Uozumi, J.

A. Dogariu, J. Uozumi, and T. Asakura, “Wavelet transform analysis of slightly rough surfaces,” Opt. Commun. 107, 1–5(1994).
[CrossRef]

Urban, K.

K. Bittner and K. Urban, “Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions,” Appl. Comput. Harmon. Anal. 24, 94–119(2008).
[CrossRef]

P. Jester, C. Menke, and K. Urban, “Wavelet methods for the representation, analysis and simulation of optical surfaces,” submitted to IMA J. Appl. Math.

K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University, 2009).

Vogl, T. P.

Yamagata, Y.

Appl. Comput. Harmon. Anal. (1)

K. Bittner and K. Urban, “Adaptive wavelet methods using semiorthogonal spline wavelets: Sparse evaluation of nonlinear functions,” Appl. Comput. Harmon. Anal. 24, 94–119(2008).
[CrossRef]

Appl. Opt. (4)

Commun. Pure Appl. Math. (1)

A. Cohen, I. Daubechies, and J.-C. Feauveau, “Biorthogonal bases of compactly supported wavelets,” Commun. Pure Appl. Math. 45, 485–560 (1992).
[CrossRef]

Meas. Sci. Technol. (1)

C.-L. Tien and Y.-R. Lyu, “Optical surface flatness recognized by discrete wavelet transform and grey level co-occurrence matrix,” Meas. Sci. Technol. 17, 2299–2305 (2006).
[CrossRef]

Opt. Commun. (1)

A. Dogariu, J. Uozumi, and T. Asakura, “Wavelet transform analysis of slightly rough surfaces,” Opt. Commun. 107, 1–5(1994).
[CrossRef]

Opt. Eng. (1)

S. A. Lerner and J. M. Sasian, “Use of implicitly defined optical surfaces for the design of imaging and illumination systems,” Opt. Eng. 39, 1796–1801 (2000).
[CrossRef]

Opt. Express (2)

Proc. SPIE (4)

A. W. Greynolds, “Superconic and subconic surface descriptions in optical design,” Proc. SPIE 4832, 1–9 (2002).
[CrossRef]

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

P. Ott, “Optic design of head-up displays with freeform surfaces specified by NURBS,” Proc. SPIE 7100, 71000Y (2008).
[CrossRef]

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

Other (9)

C. de Boor, A Practical Guide to Splines, revised ed., Vol.  27 of Applied Mathematical Sciences (Springer-Verlag, 2001).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

P. Jester, C. Menke, and K. Urban, “Wavelet methods for the representation, analysis and simulation of optical surfaces,” submitted to IMA J. Appl. Math.

K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University, 2009).

R. A. Adams, Sobolev Spaces, Vol.  65 of Pure and Applied Mathematics (Academic, 1975).

K. P. Thompson, F. Fournier, J. P. Rolland, and G. W. Forbes, “The Forbes polynomial: a more predictable surface for fabricators,” in Optical Fabrication and Testing (Optical Society of America, 2010), paper OTuA6.

Flexible Library for Efficient Numerical Solutions and Library for Adaptive Wavelet Applications, Institute for Numerical Mathematics, Ulm University.

Optische Analyse und Synthese, Carl Zeiss AG.

P. Jester, “Beschreibung optischer Grenzflächen mit Wavelets,” Diploma thesis, Institute for Numerical Mathematics (Ulm University, Germany, 2007).

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Figures (6)

Fig. 1
Fig. 1

Contour-color plot of the test surfaces (a) KXY, (b)  KSA 1 , and (c)  KSA 2 .

Fig. 2
Fig. 2

RMS error of the approximation of surface KXY.

Fig. 3
Fig. 3

RMS error of the approximation of the partial derivative x P j f KSA 1 .

Fig. 4
Fig. 4

RMS error of the approximation of surface KSA 2 , with undefined values in the bounding box.

Fig. 5
Fig. 5

Maximal deviation M x , j of the intersection point.

Fig. 6
Fig. 6

Maximal deviation of the normal M n , j of the intersection point.

Tables (2)

Tables Icon

Table 1 Parameters of the Test Surfaces KXY, KSA 1 , and KSA 2

Tables Icon

Table 2 Computing Time in Seconds for the Evaluation of the B-Spline Representation

Equations (24)

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N 1 ( x ) { 1 , if     x [ 0 , 1 ) , 0 , else, N d ( x ) 0 1 N d 1 ( x t ) d t = ( N d 1 * N 1 ) ( x ) ,
supp N d [ 0 , d ] ,
N d ( x ) = x d 1 N d 1 ( x ) + d x d 1 N d 1 ( x 1 ) ,
N d C d 2 ( R ) with N d ( x ) = N d 1 ( x ) N d 1 ( x 1 ) .
N d ( x ) = 2 1 d k = 0 d ( d k ) N d ( 2 x k ) , x R .
P j f k c j , k φ j , k , c j , k ( f , φ ˜ j , k ) L 2 ( Ω ) ,
f P j f L 2 ( Ω ) = O ( 2 j s ) , 0 s d ,
c j , k c ¯ j , k 2 j 2 = m m γ d , f ( 2 j ( k + ) ) ,
P j qi f k c ¯ j , k φ j , k ,
f P j qi f L 2 ( Ω ) = O ( 2 j s ) , 0 s d ,
f KXY ( x , y ) ρ x x 2 + ρ y y 2 1 + [ 1 ( 1 + κ x ) ( ρ x x ) 2 ( 1 + κ y ) ( ρ y y ) 2 ] 1 / 2 + c 1 x + c 2 y + c 3 x 2 + c 4 x y + c 5 y 2 + c 6 x 3 + .
f KSA ( r ) ρ r 2 1 + [ 1 ( 1 + κ ) ( ρ r ) 2 ] 1 / 2 + k = 2 c k 1 r 2 k , where     r 2 x 2 + y 2 .
err j ( f ) RMS [ ( f P j qi f ) ( x i 1 , y i 2 ) ] .
r ( α ) p + α d , α R ,
n = n ( x , y ) = e 1 × e 2 = ( x g ( x , y ) , y g ( x , y ) , 1 ) T ,
F ( α ) p z + α d z g ( p x + α d x , p y + α d y ) = 0 , F :     R R .
F ( α ) = d z x g ( p x + α d x , p y + α d y ) d x y g ( p x + α d x , p y + α d y ) d y = n ( x ( α ) , y ( α ) ) T · d ,
cos ( γ ) = [ 1 ( n n ) 2 ( 1 cos 2 ( γ ) ) ] 1 2 ,
M x , j max i = 1 , , n { x i ( O ) x i ( j ) } , M n , j max i = 1 , , n { n i ( O ) n i ( j ) } ,
ψ ( x ) = k Z b k φ ( 2 x k ) ,
ψ ( x ) { 1 , if     x [ 0 , 1 2 ) , 1 if     x [ 1 2 , 1 ) , 0 else .
FWT :     ( c j ) ( c 0 , d 0 , , d j 1 ) ,
P j f = k Z c 0 , k φ 0 , k + = 0 j 1 k Z d , k ψ , k ,
c j 1 , k = 1 2 m c j , m a m 2 k , d j 1 , k = 1 2 m c j , m b m 2 k ,

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