Abstract

A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.

© 2016 Optical Society of America

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References

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  1. R. N. Wilson, Reflecting Telescope Optics I, (Springer, 2004).
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  5. D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin Exp Optom. 91, 251–264 (2008).
    [Crossref] [PubMed]
  6. R. Navarro, “Adaptive model of the aging emmetropic eye and its changes with accommodation,” J. Vis. 14(13), 21 (2014).
    [Crossref]
  7. F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Engineering 39, 10–22 (2000).
    [Crossref]
  8. D. Michaelis, P. Schreiber, and A. Brüuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36, 918–920 (2011).
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    [Crossref]
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  20. E. W. Weisstein, “Associated Legendre Polynomial”, From MathWorld. A Wolfram Web Resource. http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html .

2015 (1)

2014 (2)

2013 (1)

2012 (1)

2011 (2)

2010 (1)

2009 (2)

2008 (2)

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin Exp Optom. 91, 251–264 (2008).
[Crossref] [PubMed]

Y. Ding, X. Liu, Z.-R. Zheng, and P.-F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16, 12958–12966 (2008).
[Crossref] [PubMed]

2007 (1)

2002 (1)

2000 (1)

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Engineering 39, 10–22 (2000).
[Crossref]

1954 (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[Crossref]

Antón, I.

Bhatia, A. B.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[Crossref]

Born, M.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[Crossref]

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Engineering 39, 10–22 (2000).
[Crossref]

Brüuer, A.

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Engineering 39, 10–22 (2000).
[Crossref]

Cobb, M. J.

Díaz, J. A.

Ding, Y.

Domínguez, C.

Ferreira, C.

Fisher, S. W.

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin Exp Optom. 91, 251–264 (2008).
[Crossref] [PubMed]

Forbes, G. W.

Gu, P.-F.

Li, X.

Liu, X.

Lopéz, J. L.

López, J. L.

Meister, D. J.

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin Exp Optom. 91, 251–264 (2008).
[Crossref] [PubMed]

Michaelis, D.

Muschaweck, J.

Navarro, R.

Pérez Sinusía, E.

Ries, H.

Sala, G.

Schreiber, P.

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Engineering 39, 10–22 (2000).
[Crossref]

Victoria, M.

Wilson, R. N.

R. N. Wilson, Reflecting Telescope Optics I, (Springer, 2004).

Wolf, E.

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[Crossref]

Wu, Y.

Xi, J.

Zheng, Z.-R.

Appl. Opt. (1)

Clin Exp Optom. (1)

D. J. Meister and S. W. Fisher, “Progress in the spectacle correction of presbyopia. Part 2: Modern progressive lens technologies,” Clin Exp Optom. 91, 251–264 (2008).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

J. Vis. (1)

R. Navarro, “Adaptive model of the aging emmetropic eye and its changes with accommodation,” J. Vis. 14(13), 21 (2014).
[Crossref]

Opt. Engineering (1)

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Engineering 39, 10–22 (2000).
[Crossref]

Opt. Express (8)

Opt. Lett. (2)

Proc. Camb. Philos. Soc. (1)

A. B. Bhatia, E. Wolf, and M. Born, “On the circle polynomials of Zernike and related orthogonal sets,” Proc. Camb. Philos. Soc. 50(1), 40–48 (1954).
[Crossref]

Other (4)

E. W. Weisstein, “Legendre Polynomial”, From MathWorld. A Wolfram Web Resource. http://mathworld.wolfram.com/LegendrePolynomial.html .

E. W. Weisstein, “Spherical Harmonic”, From MathWorld. A Wolfram Web Resource. http://mathworld.wolfram.com/SphericalHarmonic.html .

E. W. Weisstein, “Associated Legendre Polynomial”, From MathWorld. A Wolfram Web Resource. http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html .

R. N. Wilson, Reflecting Telescope Optics I, (Springer, 2004).

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Figures (7)

Fig. 1
Fig. 1 Graphs of the first five functions of the new quasi-orthonormal basis {qn (r)} n =0,1,2,… (see Eq. (18)) obtained from the normalized Legendre polynomials and q 0 ( r ) = b 1 s L 2 r 2 a 2 , with s = 1, b = 1 and a = L = 3/4 using different scale: q 0 (orange), q 1 (red), q 2 (blue), q 3 (green), q 4 (brown).
Fig. 2
Fig. 2 First functions q n m ( r , θ ) (see Eq. (34)) for the case of the spherical harmonics and q 0 0 ( r ) = 1 r 2 / 4 The rows represent the ascending order from n = 0 to n = 4, the columns are the positive values of m from m = 0 to m = n.
Fig. 3
Fig. 3 Gaussian surface of revolution Eq. (35) given in Example 1.
Fig. 4
Fig. 4 Plots of the fit error obtained using our basis functions and Forbes polynomials, Q m con and Q m bfs , for the same number of terms, eight, for the three cases.
Fig. 5
Fig. 5 Relative errors in the approximation of the function Eq. (35) using our approximation (see Eq. (19)) and Forbes’ approximations (see Eqs. (36) and (37)) in the L 2−norm for the same number of terms.
Fig. 6
Fig. 6 Non-symmetric elliptical gaussian surface given in Eq. (41).
Fig. 7
Fig. 7 RMSE in the approximation of the function Eq. (41) provided by the different approximations using the number of terms.

Tables (1)

Tables Icon

Table 1 First five functions of the new quasi-orthonormal basis {qn (r)} n =0,1,2,… (see Eq. (18)) obtained from the normalized Legendre polynomials and q 0 ( r ) = b 1 s L 2 r 2 a 2 .

Equations (42)

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δ m , n = c d p n ( x ) p m ( x ) ρ ( x ) d x , n , m = 0 , 1 , 2 ,
δ m , n = 0 1 p n ( φ ( r ) ) p m ( φ ( r ) ) ρ ( r ) ) φ ( r ) d r , n , m = 0 , 1 , 2 ,
q n ( r ) : = C n p n ( φ ( r ) ) w ( r ) , w ( r ) : = ρ ( φ ( r ) ) φ ( r ) r ,
{ ρ ( φ ( r ) ) φ ( r ) = 1 C 0 2 p 0 2 r q 0 2 ( r ) , φ ( 0 ) = c , φ ( 1 ) = d .
c x ρ ( t ) d t = 1 C 0 2 p 0 2 0 r s q 0 2 ( s ) d s ,
C 0 2 = 0 1 s q 0 2 ( s ) d s / [ p 0 2 c d ρ ( t ) d t ] .
q n ( r ) = q 0 ( r ) C 0 p 0 C n p n ( φ ( r ) ) , n = 0 , 1 , 2 ,
f ( x ) : = F ( φ 1 ( x ) ) w ( φ 1 ( x ) ) ,
f ρ 2 = a b | f ( x ) | 2 ρ ( x ) d x = 0 1 | f ( φ ( r ) ) | 2 ρ ( φ ( r ) ) φ ( r ) d r = 0 1 | F ( r ) | 2 r d r = F r 2 .
f ( x ) = n = 0 c n p n ( x ) , almost everywhere in [ c , d ] ,
c n : = c d p n ( x ) f ( x ) ρ ( x ) d x = 1 C n 0 1 q n ( r ) F ( r ) r d r ; n = 0 , 1 , 2 , 3 ,
F ( r ) = w ( r ) f ( φ ( r ) ) = n = 0 c n w ( r ) p n ( φ ( r ) ) = c 0 C 0 q 0 ( r ) + n = 1 c n q n ( r ) .
p n ( x ) = 2 n + 1 2 1 2 n k = 0 n ( n k ) 2 ( x 1 ) n k ( x + 1 ) k , p 0 ( x ) = 1 2 ,
q 0 ( r ) = b 1 s L 2 r 2 a 2 ,
x + 1 = 1 x d t = 2 C 0 2 0 r t ( b 2 s b 2 a 2 L 2 t 2 ) d t = b 2 r 2 C 0 2 [ 1 s L 2 a 2 r 2 2 ] .
C 0 = 1 2 2 b 2 s b 2 a 2 L 2 .
x = φ ( r ) = 2 r 2 2 a 2 s L 2 [ 2 a 2 s L 2 r 2 ] 1.
q n ( r ) = 2 C n sign ( b ) 2 a 2 2 s L 2 r 2 2 a 2 s L 2 p n ( 2 r 2 2 a 2 s L 2 [ 2 a 2 s L 2 r 2 ] 1 ) , n = 0 , 1 , 2 , ,
F ( r ) = c 0 C 0 b 1 s L 2 r 2 a 2 + n = 1 c n q n ( r ) ,
q 0 ( r ) = c L 2 r 2 1 + 1 ε c 2 L 2 r 2 ,
q n ( r ) = 2 q 0 ( r ) C 0 C n p n ( φ ( r ) ) , n = 0 , 1 , 2 , ,
C 0 2 = 8 [ ( 1 ε c 2 L 2 ) 3 / 2 1 ] + 3 ε c 2 L 2 ( 4 ε c 2 L 2 ) 12 ε 3 c 4 L 2 , C n = 1 , n = 1 , 2 , 3 ,
φ ( r ) = 2 8 [ ( 1 ε c 2 L 2 r 2 ) 3 / 2 1 ] + 3 ε c 2 L 2 r 2 ( 4 ε c 2 L 2 r 2 ) 8 [ ( 1 ε c 2 L 2 ) 3 / 2 1 ] + 3 ε c 2 L 2 ( 4 ε c 2 L 2 ) 1 ,
q n ( r ) = 2 6 C n c 3 L 3 r 2 1 + 1 ε c 2 L 2 r 2 ε 3 8 [ ( 1 ε c 2 L 2 ) 3 / 2 1 ] + 3 ε c 2 L 2 ( 4 ε c 2 L 2 ) × p n ( 2 8 [ ( 1 ε c 2 L 2 r 2 ) 3 / 2 1 ] + 3 ε c 2 L 2 r 2 ( 4 ε c 2 L 2 r 2 ) 8 [ ( 1 ε c 2 L 2 ) 3 / 2 1 ] + 3 ε c 2 L 2 ( 4 ε c 2 L 2 ) 1 ) , n = 0 , 1 , 2 ,
δ n , n δ m , m = c d ρ ( x ) d x 0 2 π d θ p n m ( x , θ ) p n m ( x , θ ) , n , m ( n ) = 0 , 1 , 2 ,
δ n , n δ m , m = 0 1 ρ ( φ ( r ) ) φ ( r ) d r 0 2 π d θ p n m ( φ ( r ) , θ ) p n m ( φ ( r ) , θ ) , n , m ( n ) = 0 , 1 , 2 ,
q n m ( r , θ ) : = C n m p n m ( φ ( r ) , θ ) w ( r ) , w ( r ) : = ρ ( φ ( r ) ) φ ( r ) r ,
q n m ( r , θ ) = C n m w ( r ) p n m ( φ ( r ) , θ ) ,
( C 0 0 ) 2 = 0 1 s ( q 0 0 ( s ) ) 2 d s / [ ( p 0 0 ) 2 c d ρ ( t ) d t ] .
F ( r , θ ) = c 0 0 C 0 0 q 0 0 ( r ) + n , m ( n ) = 0 ( n , m ) ( 0 , 0 ) c n m q n m ( r , θ ) ,
c n m : 1 C n m D q n m ( r , θ ) F ( r , θ ) r d r d θ , n , m ( n ) = 0 , 1 , 2 , 3 ,
p n m ( x , θ ) = ( 2 δ m , 0 ) ( 2 n + 1 ) ( n m ) ! 4 π ( n + m ) ! { P n m ( x ) cos ( m θ ) , 0 m n , P n m ( x ) sin ( m θ ) , n m < 0 ,
P n m ( x ) = ( 1 ) m 2 n n ! ( 1 x 2 ) m / 2 d n + m d x n + m ( x 2 1 ) n , p 0 0 ( x , θ ) = 1 2 π .
q n m ( r , θ ) = 2 C n sign ( b ) 2 a 2 2 s L 2 r 2 2 a 2 s L 2 p n m ( 2 r 2 2 a 2 s L 2 [ 2 a 2 s L 2 r 2 ] 1 , θ )
f ( r ) = 1 2 π ( e r 2 2 1 ) , r [ 0 , 1 ] .
F ( r ) c r 2 1 + 1 ε c 2 r 2 + u 4 m = 0 M a m Q m con ( u 2 ) ,
F ( r ) c bfs r 2 1 + 1 c bfs 2 r 2 + u 2 ( 1 u 2 ) 1 c bfs 2 r 2 m = 0 M a m Q m bfs ( u 2 ) ,
f ( r , θ ) = 1 2 π ( e r 2 2 1 ) , r [ 0 , 1 ] , θ ( 0 , 2 π ] .
f ( r , θ ) c 0 C 0 ( c bfs ) 2 r 2 + n , m ( n ) = 0 ( n , m ) ( 0 , 0 ) N c n m q n m ( r , θ ) ,
f ( r , θ ) n = 0 N m = n n c ˜ n m Z n m ( r , θ ) , c ˜ n m = 1 π 0 2 π 0 1 Z n m ( r , θ ) f ( r , θ ) r d r d θ ,
f ( r , θ ) = 2 e r 2 ( cos 2 θ + 2 sin 2 θ ) , r [ 0 , 1 ] , θ [ 0 , 2 π ) .
f ( r , θ ) c r 2 1 + 1 c 2 r 2 + 1 1 c 2 r 2 { u 2 ( 1 u 2 ) n = 0 N a n 0 Q n 0 ( u 2 ) + m = 1 M u m n = 0 N [ a n m cos m θ + b n m sin m θ ] Q n m ( u 2 ) } , u = r r m a x ,

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