Abstract

Advances in fabrication and testing are allowing aspheric optics to have greater impact through their increased prevalence and complexity. The most widely used characterization of surface shape is numerically deficient, however. Furthermore, with regard to tolerancing and to constraints for manufacturability, this representation is poorly suited for design purposes. Effective alternatives are therefore presented for working with rotationally symmetric surfaces that are either (i) strongly aspheric or (ii) constrained in terms of the slope in the departure from a best-fit sphere.

© 2007 Optical Society of America

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References

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  1. See, for example, the discussion and references in H. Chase, "Optical design with rotationally symmetric NURBS", SPIE Proceedings 4832, 10-24 (2002) and A. W. Greynolds, "Superconic and subconic surface descriptions in optical design," Proc. SPIE 4832, 1-9 (2002). Such matters are also treated within the manuals for commercial optical design software.
    [CrossRef]
  2. G. H. Spencer and M. V. R.K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962), see Eq. (16).
    [CrossRef]
  3. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), see 22.2.1.
  4. E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. A: Math. Gen. 35, 3467-3478 (2002).
    [CrossRef]
  5. E. W. Weisstein, "Jacobi Polynomial" from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html, see esp. Eqs. (10-14).
  6. B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal.  1, 229-234 (1981).
    [CrossRef]
  7. A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.)
    [CrossRef]

2006 (1)

A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.)
[CrossRef]

2002 (2)

See, for example, the discussion and references in H. Chase, "Optical design with rotationally symmetric NURBS", SPIE Proceedings 4832, 10-24 (2002) and A. W. Greynolds, "Superconic and subconic surface descriptions in optical design," Proc. SPIE 4832, 1-9 (2002). Such matters are also treated within the manuals for commercial optical design software.
[CrossRef]

E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. A: Math. Gen. 35, 3467-3478 (2002).
[CrossRef]

1981 (1)

B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal.  1, 229-234 (1981).
[CrossRef]

Dirksen, P.

A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.)
[CrossRef]

Doha, E. H.

E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. A: Math. Gen. 35, 3467-3478 (2002).
[CrossRef]

Janssen, A. J. E. M.

A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.)
[CrossRef]

Luke, Y. L.

B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal.  1, 229-234 (1981).
[CrossRef]

Murty, M. V. R.K.

G. H. Spencer and M. V. R.K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962), see Eq. (16).
[CrossRef]

Spencer, G. H.

G. H. Spencer and M. V. R.K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962), see Eq. (16).
[CrossRef]

Ting, B. Y.

B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal.  1, 229-234 (1981).
[CrossRef]

IMA J. Numer. Anal. (1)

B. Y. Ting and Y. L. Luke, "Conversion of polynomials between different polynomial bases," IMA J. Numer. Anal.  1, 229-234 (1981).
[CrossRef]

J. Phys. A: Math. Gen. (1)

E. H. Doha, "On the coefficients of differentiated expansions and derivatives of Jacobi polynomials," J. Phys. A: Math. Gen. 35, 3467-3478 (2002).
[CrossRef]

Microsyst. (1)

A. J. E. M. Janssen and P. Dirksen, "Concise formula for the Zernike coefficients of scaled pupils," J. Microlithogr. Microfabr. Microsyst. 5, 1-3 (2006). (Note that the Zernikes are also Jacobi polynomials.)
[CrossRef]

Proc. SPIE (1)

See, for example, the discussion and references in H. Chase, "Optical design with rotationally symmetric NURBS", SPIE Proceedings 4832, 10-24 (2002) and A. W. Greynolds, "Superconic and subconic surface descriptions in optical design," Proc. SPIE 4832, 1-9 (2002). Such matters are also treated within the manuals for commercial optical design software.
[CrossRef]

Other (3)

G. H. Spencer and M. V. R.K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962), see Eq. (16).
[CrossRef]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), see 22.2.1.

E. W. Weisstein, "Jacobi Polynomial" from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html, see esp. Eqs. (10-14).

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Figures (5)

Fig. 1.
Fig. 1.

Plots of the orthogonal basis elements for m = 0, 1, 2,… 5.

Fig. 2.
Fig. 2.

Plots for m = 0, 1, 2,… 5 of the basis elements that are tailored for use when constraining the departure from a best-fitting sphere along its normal.

Fig. 3.
Fig. 3.

Plots of the orthogonal slope functions of Eq. (12) for m = 0, 1, 2,… 5.

Fig. 4.
Fig. 4.

The Cartesian oval used as an example. For some purposes, it can be helpful to note that a parametric description as η(θ) follows upon solving a quadratic.

Fig. 5.
Fig. 5.

Plots of (a) the Cartesian oval’s departure from the best-fitting conic, and (b) the error in the associated polynomial fit with M=7.

Tables (1)

Tables Icon

Table 1. Fit coefficients (in nanometer units) for two different bases. [To clarify a point made in the text, the central column contains just the monomial expansion of Q con 7(x), c.f. Eq. (9).]

Equations (22)

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

z ( ρ ) = c ρ 2 ( 1 + 1 ε c 2 ρ 2 ) + m = 0 M a m ρ 2 m + 4 ,
z ( ρ ) = c ρ 2 ( 1 + 1 ε c 2 ρ 2 ) + D con ( ρ ρ max ) ,
D con ( u ) u 4 m = 0 M a m Q m con ( u ) 2 .
E 2 ( a 0 , a 1 , a M ) [ g ( u ρ max ) u 4 m = 0 M a m Q m con ( u ) 2 ] 2 .
h ( u ) 0 1 h ( u ) w ( u 2 ) u du 0 1 w ( v 2 ) v dv .
n = 0 M G m n a n = b m ,
G m n u 8 Q m con ( u 2 ) Q n con ( u 2 ) = 0 1 Q m con ( x ) Q n con ( x ) x 4 d x .
Q m con ( x ) P m ( 0,4 ) ( 2 x 1 ) .
Q 0 con ( x ) = 1 , Q 1 con ( x ) = ( 5 6 x ) , Q 2 con ( x ) = 15 14 x ( 3 2 x ) ,
Q 3 con ( x ) = { 35 12 x [ 14 x ( 21 10 x ) ] } ,
Q 4 con ( x ) = 70 3 x { 168 5 x [ 84 11 x ( 8 3 x ) ] } ,
Q 5 con ( x ) = [ 126 x ( 1260 11 x { 420 x [ 720 13 x ( 45 14 x ) ] } ) ] .
z ( ρ ) = c bfs ρ 2 ( 1+ 1 c bfs 2 ρ 2 ) + D bfs ( ρ ρ max ) ,
D bfs ( u ) u 2 ( 1 u 2 ) 1 c bfs 2 ρ max 2 u 2 m = 0 M a m Q m bfs ( u 2 ) .
S m ( u ) d d u { u 2 ( 1 u 2 ) Q m bfs ( u 2 ) } ,
S m ( u ) S n ( u ) = δ mn ,
Q 0 bfs ( x ) = 1 , Q 1 bfs ( x ) = 1 19 ( 13 16 x ) , Q 2 bfs ( x ) = 2 95 [ 29 4 x ( 25 19 x ) ] ,
Q 3 bfs ( x ) = 2 2545 { 207 4 x [ 315 x ( 577 320 x ) ] } ,
Q 4 bfs ( x ) = 1 131831 3 ( 7737 16 x { 4653 2 x [ 7381 8 x ( 1168 509 x ) ] } ) ,
Q 5 bfs ( x ) = 1 6632213 3 [ 66657 32 x ( 28338 x { 135325 8 x [ 35884 x ( 34661 12432 x ) ] } ) ] .
{ 1 ρ max d d u [ u 2 ( 1 u 2 ) m = 0 M a m Q m bfs ( u 2 ) ] } 2 = { 1 ρ max m = 0 M a m S m ( u ) } 2 = 1 ρ max 2 m = 0 M a m 2 .
m = 0 M a m 2 < ( γ N λ 8 ) 2 .

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