Abstract

Whether in design or the various stages of fabrication and testing, an effective representation of an asphere’s shape is critical. Some algorithms are given for implementing tailored polynomials that are ideally suited to these needs. With minimal coding, these results allow a recently introduced orthogonal polynomial basis to be employed to arbitrary orders. Interestingly, these robust and efficient methods are enabled by the introduction of an auxiliary polynomial basis.

© 2010 OSA

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  1. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15, 5218–5226, (2007) http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218.
    [Crossref] [PubMed]
  2. G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics”, SPIE Optifab (2009) TD06–25 (1). Available at http://www.qedmrf.com.
  3. C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004), doi:10.1117/12.551420.
    [Crossref]
  4. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862, (2010) http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13851.
    [Crossref] [PubMed]
  5. E. W. Weisstein, “Jacobi Polynomial” from MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/JacobiPolynomial.html, see esp. Equations (10–14).
  6. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.
  7. C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.
  8. F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19, 33–36 (1965).
    [Crossref]
  9. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.
  10. A useful overview is given at http://en.wikipedia.org/wiki/Discrete_cosine_transform.
  11. G. W. Forbes, “Can you make/measure this asphere for me?”, Frontiers in Optics, OSA Technical Digest (2009), http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThH1.

2010 (1)

2007 (1)

2004 (1)

C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004), doi:10.1117/12.551420.
[Crossref]

1965 (1)

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19, 33–36 (1965).
[Crossref]

1955 (1)

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.

Brophy, C. P.

G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics”, SPIE Optifab (2009) TD06–25 (1). Available at http://www.qedmrf.com.

Clenshaw, C. W.

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.

Forbes, G. W.

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862, (2010) http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13851.
[Crossref] [PubMed]

G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Express 15, 5218–5226, (2007) http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-8-5218.
[Crossref] [PubMed]

G. W. Forbes, “Can you make/measure this asphere for me?”, Frontiers in Optics, OSA Technical Digest (2009), http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThH1.

G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics”, SPIE Optifab (2009) TD06–25 (1). Available at http://www.qedmrf.com.

Jeu, C. du

C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004), doi:10.1117/12.551420.
[Crossref]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.

Smith, F. J.

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19, 33–36 (1965).
[Crossref]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.

Weisstein, E. W.

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

Math. Comput. (1)

F. J. Smith, “An Algorithm for Summing Orthogonal Polynomial Series and their Derivatives with Applications to Curve-Fitting and Interpolation,” Math. Comput. 19, 33–36 (1965).
[Crossref]

Math. Tables Other Aids Comput. (1)

C. W. Clenshaw, “A Note on the Summation of Chebyshev Series,” Math. Tables Other Aids Comput. 9, 118–120 (1955), http://www.jstor.org/stable/2002068.

Opt. Express (2)

Proc. SPIE (1)

C. du Jeu, “Criterion to appreciate difficulties of aspherical polishing,” Proc. SPIE 5494, 113–121 (2004), doi:10.1117/12.551420.
[Crossref]

Other (6)

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

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1978), Chap. 22.

G. W. Forbes and C. P. Brophy, “Designing cost-effective systems that incorporate high-precision aspheric optics”, SPIE Optifab (2009) TD06–25 (1). Available at http://www.qedmrf.com.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 1992) Section 2.9.

A useful overview is given at http://en.wikipedia.org/wiki/Discrete_cosine_transform.

G. W. Forbes, “Can you make/measure this asphere for me?”, Frontiers in Optics, OSA Technical Digest (2009), http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2009-FThH1.

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Figures (5)

Fig. 1.
Fig. 1.

A sample of the slope-orthogonal polynomials used in this work. By construction, they are not only orthogonal in slope, but normalized so that their mean square slope is unity.

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Fig. 2.
Fig. 2.

A sample of the scaled Jacobi polynomials used in this work. The plot at left is for comparison with Fig. 1, and the one on the right holds up to m = 20 to reveal their envelope.

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Fig. 3.
Fig. 3.

A cross-section of the parabola used for demonstration (in blue) is shown with its bestfit sphere (dotted red). The aspheric departure (viz. sag difference) is plotted at right.

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Fig. 4.
Fig. 4.

Log plot of {bm } in nm units. The dots are red/blue for positive/negative values. The 1nm reference level is drawn as a solid gray line. The plot at right is the error in the fit that results if all but the last red dot above the 1nm line are retained (i.e. m = 0 to 6 are kept).

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Fig. 5.
Fig. 5.

The first ten basis members and their slopes for an obstructed aperture with ε = 0.25.

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Equations (70)

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z = c ρ 2 1 + 1 − c 2 ρ 2 + u 2 ( 1 − u 2 ) 1 − c 2 ρ 2 ∑ m = 0 M a m Q m ( u 2 ) ,
( 2 π ) ∫ 0 1 S m ( u ) S n ( u ) 1 1 − u 2 d u = δ mn ,
S m ( u ) ≔ d d u [ u 2 ( 1 − u 2 ) Q m ( u 2 ) ] .
∫ 0 1 1 1 − u 2 d u = π 2 .
P m ( x ) = f m Q m ( x ) + g m − 1 Q m − 1 ( x ) + h m − 2 Q m − 2 ( x ) ,
P m + 1 ( x ) = ( 2 − 4 x ) P m ( x ) − P m − 1 ( x ) .
Q m + 1 ( x ) = [ P m + 1 ( x ) − g m Q m ( x ) − h m − 1 Q m − 1 ( x ) ] f m + 1 ,
S ( x ) ≔ ∑ m = 0 M a m Q m ( x ) .
a m = f m b m + g m b m + 1 + h m b m + 2 ,
a M − 1 = f M − 1 b M − 1 + g M − 1 b M and a M = f M b M .
b M = a M f M and b M − 1 = ( a M − 1 − g M − 1 b M ) f M − 1 ,
b m = ( a m − g m b m + 1 − h m b m + 2 ) f m .
S ( x ) ≡ ∑ m = 0 M b m P m ( x ) ,
α M = b M and α M − 1 = b M − 1 + ( 2 − 4 x ) α M ,
α m = b m + ( 2 − 4 x ) α m + 1 − α m + 2 ,
S ( x ) = [ α 0 − ( 2 − 4 x ) α 1 ] P 0 ( x ) + α 1 P 1 ( x )
= 2 ( α 0 + α 1 ) .
α M − j + 1 ( j ) = 0 and α M − j ( j ) = − 4 j α M − j + 1 ( j − 1 ) ,
α m ( j ) = ( 2 − 4 x ) α m + 1 ( j ) − α m + 2 ( j ) − 4 j α m + 1 ( j − 1 ) .
S ( j ) ( x ) = 2 ( α 0 ( j ) + α 1 ( j ) ) .
z ( ρ ) = c ρ 2 1 + φ + u 2 ( 1 − u 2 ) φ S ( u 2 ) ,
z ′ ( ρ ) = c ρ φ + u [ 1 + φ 2 − u 2 ( 1 + 3 φ 2 ) ] ρ max φ 3 S ( u 2 ) + 2 u 3 ( 1 − u 2 ) ρ max φ S ′ ( u 2 ) ,
z ″ ( ρ ) = c φ 3 + 3 − φ 2 − 3 u 2 ( 1 + φ 2 + 2 φ 4 ) ρ max 2 φ 5 S ( u 2 )
+ 2 u 2 [ 2 + 3 φ 2 − u 2 ( 2 + 7 φ 2 ) ] ρ max 2 φ 3 S ′ ( u 2 ) + 4 u 4 ( 1 − u 2 ) ρ max 2 φ S ″ ( u 2 ) .
c axial = z ″ ( 0 ) = c + 2 ρ max 2 S ( 0 ) = c + 4 ρ max 2 ∑ m = 0 M ( 2 m + 1 ) b m .
z = f ( ρ ) ,
c = 2 f ( ρ max ) ρ max 2 + f ( ρ max ) 2 .
f ( ρ ) ≈ c ρ 2 1 + 1 − c 2 ρ 2 + u 2 ( 1 − u 2 ) 1 − c 2 ρ 2 ∑ m = 0 M a m Q m ( u 2 ) .
∑ m = 0 M b m P m ( u 2 ) ≈ 1 − ( u c ρ max ) 2 u 2 ( 1 − u 2 ) { f ( u ρ max ) − c ( u ρ max ) 2 1 + 1 − ( u c ρ max ) 2 } ≕ F ( u ) .
b m ≈ ( − 1 ) m 2 N ∑ j = 0 N − 1 F j cos [ π N ( m + 1 2 ) ( j + 1 2 ) ] ,
F j ≔ cos ( π 2 N ( j + 1 2 ) ) F [ cos ( π 2 N ( j + 1 2 ) ) ] .
b = { 1009010.04959 , 2770.64974485 , − 4739.30847163
1172.09704743 , − 257.270488293 , 55.4172061289 ,
− 11.966650385 , 2.604636 675 85 } nm .
a = { 2019004 , 7143 , − 13944 , 4190 , − 1095 , 283 , − 68 } nm .
z = c ρ 2 1 + 1 − c 2 ρ 2 + ( u 2 − ε 2 ) ( 1 − u 2 ) ( 1 − ε 2 ) ( 1 + ε ) ( 1 − c 2 ρ 2 ) ∑ m = 0 M a m Q m ( u 2 − ε 2 1 − ε 2 ) ,
[ 2 ( 1 − ε ) π ] ∫ ε 1 S m ( u ) S n ( u ) 1 u u 2 − ε 2 1 − u 2 d u ,
S m ( u ) ≔ 1 ( 1 − ε 2 ) 1 + ε d d u [ ( u 2 − ε 2 ) ( 1 − u 2 ) Q m ( u 2 − ε 2 1 − ε 2 ) ] .
∫ ε 1 1 u u 2 − ε 2 1 − u 2 d u = ( 1 − ε ) π 2 .
( 2 π ) ∫ 0 1 { [ ( 1 − 2 x ) Q m ( x ) + x ( 1 − x ) Q m ′ ( x ) ]
× [ ( 1 − 2 x ) Q n ( x ) + x ( 1 − x ) Q n ′ ( x ) ] } 4 x 1 − x d x = δ mn ,
∫ 0 1 φ n ( x ) P m ( α , β ) ( 2 x − 1 ) ( 1 − x ) α x β d x = 0 , for m > n ,
d d x [ P n ( α , β ) ( x ) ] = 1 2 ( n + α + β + 1 ) P n − 1 ( α + 1 , β + 1 ) ( x ) .
P m ( x ) ≔ ( − 1 ) m 2 ( 2 m ) !! ( 2 m − 1 ) !! P m ( − 1 2 , 1 2 ) ( 2 x − 1 ) ,
〈 f , g 〉 ≔ ( 2 π ) ∫ 0 1 d d x [ x ( 1 − x ) f ( x ) ] d d x [ x ( 1 − x ) g ( x ) ] 4 x 1 − x d x .
P m ( x ) = ( − 1 ) m 2 cos [ 2 m + 1 2 arccos ( 2 x − 1 ) ] x ,
〈 P m , P n 〉 = ( 2 π ) ∫ 0 π 2 T m ( θ ) T n ( θ ) d θ = ( 1 π ) ∫ − π 2 π 2 T m ( θ ) T n ( θ ) d θ ,
T m ( θ ) ≔ ( − 1 ) m 2 sin θ d d θ { cos θ sin 2 θ cos [ ( 2 m + 1 ) θ ] }
= ( − 1 ) m { ( m + 2 ) cos [ ( 2 m + 3 ) θ ] + cos [ ( 2 m + 1 ) θ ] − ( m − 1 ) cos [ ( 2 m − 1 ) θ ] } .
H m = − ( 1 π ) ( m + 2 ) ( m + 1 ) ∫ − π 2 π 2 cos 2 [ ( 2 m + 3 ) θ ] d θ
= − ( m + 2 ) ( m + 1 ) 2 .
G m = − ( 1 π ) { ( m + 2 ) ∫ − π 2 π 2 cos 2 [ ( 2 m + 3 ) θ ] d θ − m ∫ − π 2 π 2 cos 2 [ ( 2 m + 1 ) θ ] d θ }
= − 1 ,
F m = ( 1 π ) [ ( m + 2 ) 2 + 1 + ( m − 1 ) 2 ] π 2
= ( m 2 + m + 3 ) , for m > 0 .
P i ( x ) = ∑ j L ij Q j ( x ) .
〈 P m , P n 〉 = 〈 ∑ j L mj Q j ( x ) , ∑ k L nk Q k ( x ) 〉
= ∑ j ∑ k L mj L nk 〈 Q j ( x ) , Q k ( x ) 〉 = ∑ j L mj L nj .
h m − 2 = H m − 2 f m − 2 = − m ( m − 1 ) ( 2 f m − 2 ) ,
g m − 1 = ( G m − 1 − g m − 2 h m − 2 ) f m − 1 = − ( 1 + g m − 2 h m − 2 ) f m − 1 ,
f m = F m − g m − 1 2 − h m − 2 2 = m ( m + 1 ) + 3 − g m − 1 2 − h m − 2 2 ,
[ f 0 0 0 0 0 0 g 0 f 1 0 0 0 0 h 0 g 1 f 2 0 0 0 0 h 1 g 2 f 3 0 0 0 0 h 2 g 3 f 4 0 0 0 0 h 3 g 4 f 5 ] = [ 2 0 0 0 0 0 − 1 2 19 4 0 0 0 0 − 1 2 − 5 2 19 4 10 19 0 0 0 0 − 6 19 − 17 2 190 1 2 509 10 0 0 0 0 − 3 2 19 10 − 91 2 5090 6 259 509 0 0 0 0 − 20 10 509 − 473 2 131831 1 2 25607 259 ] .
1 2 π ∫ 0 1 P m ( x ) P m ( x ) x 1 − x d x = δ mn .
E 2 ≔ 1 2 π ∫ 0 1 { g ( x ) − ∑ m b m P m ( x ) } 2 x 1 − x d x ,
b m = 1 2 π ∫ 0 1 g ( x ) P m ( x ) x 1 − x d x .
b m = ( − 1 ) m π ∫ − π 2 π 2 g ( cos 2 θ ) cos θ cos [ ( 2 m + 1 ) θ ] d θ .
b m = ( − 1 ) m 2 π ∫ − π 2 π 2 g ( cos 2 θ ) { cos [ 2 ( m + 1 ) θ ] + cos [ 2 m θ ] } d θ
= ( − 1 ) m 4 π ∫ 2 π g [ ( 1 + cos ψ ) 2 ] { cos [ ( m + 1 ) ψ ] + cos [ m ψ ] } d ψ .
b m ≈ ( − 1 ) m 2 N ∑ j = 0 N − 1 { 1 1 + δ j 0 g [ cos 2 ( π 2 N j ) ] cos ( π 2 N j ) } cos [ π 2 N ( 2 m + 1 ) j ] ,
b m ≈ ( − 1 ) m 2 N ∑ j = 0 N − 1 { g [ cos 2 ( π 4 N ( 2 j + 1 ) ) ] cos ( π 4 N ( 2 j + 1 ) ) } cos [ π 4 N ( 2 j + 1 ) ( 2 m + 1 ) ] .

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