Abstract

Orthogonal polynomials offer useful mathematical properties for describing freeform optical surfaces. Their advantages are best leveraged by understanding the interactions between variables such as tip and tilt, base sphere and conic variables, and packaging variables that define the problem of design for manufacture. These interactions can cause degeneracy, which can complicate the interpretation of design specifications in manufacturing and, consequently, negatively impact the cost of fabrication and assembly. Optimization constraints to break degeneracy during design are also discussed.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. F. Zernike, “Beugungstheorie des Schneidenver-Fahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
    [Crossref]
  2. V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 498–546.
  3. L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. patent4,826,287 (2May1989).
  4. T. Nakano and Y. Tamagawa, “Configuration of an off-axis three-mirror system focused on compactness and brightness,” Appl. Opt. 44, 776–783 (2005).
    [Crossref]
  5. Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
    [Crossref]
  6. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-type polynomial surfaces,” Opt. Express 19, 21919 (2011).
    [Crossref]
  7. J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
    [Crossref]
  8. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
    [Crossref]
  9. G. Forbes, “Shape specification for axially symmetric optical surfaces,” Optics Express 15, 5218–5226 (2007).
    [Crossref]
  10. G. Forbes, “Characterizing the shape of freeform optics,” Opt. Express 20, 2483–2499 (2012).
    [Crossref]
  11. C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Test. 2, 97–109 (2013).
    [Crossref]
  12. V. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [Crossref]
  13. W. Swantner and W. W. Chow, “Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes,” Appl. Opt. 33, 1832–1837 (1994).
    [Crossref]
  14. M. Bray, “Orthogonal polynomials: a set for square areas,” Proc. SPIE 5252, 314–322 (2004).
    [Crossref]
  15. D. Ochse, K. Uhlendorf, and L. Reichmann, Describing Freeform Surfaces with Orthogonal Functions (Imaging and Applied Optics, 2015).
  16. H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
    [Crossref]
  17. M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
    [Crossref]
  18. A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions—part I: mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).
    [Crossref]
  19. International Organization for Standardization, ISO 10110-19 Optics and Photonics—Preparation of Drawings for Optical Elements and Systems—Part 19 (ISO, 2015).
  20. E. W. Weisstein, “Hessian” (from MathWorld—A Wolfram Web Resource), 2018, http://mathworld.wolfram.com/Hessian.html .
  21. E. W. Weisstein, “Second derivative test” (from MathWorld—A Wolfram Web Resource), 2018, http://mathworld.wolfram.com/SecondDerivativeTest.html .
  22. J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics (Prentice-Hall, Inc., 1983).
  23. N. Takaki and J. Rolland, “Mathematical properties of describing freeform optical surfaces with orthogonal bases,” Proc. SPIE 10590, 105900U (2017).
    [Crossref]
  24. G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862 (2010).
    [Crossref]
  25. E. W. Weisstein, “Jacobi polynomials,” (from MathWorld—A Wolfram Web Resource), 2018, http://mathworld.wolfram.com/JacobiPolynomial.html .
  26. CODE V, “Zernike polynomials,” in CODE V Lens System Setup Reference Manual (2015), Appendix A.
  27. G. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862 (2010).
    [Crossref]
  28. M. Davies, Mechanical Engineering and Engineering Science, The University of North Carolina at Charlotte (personal communication, 2018).

2018 (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

2017 (3)

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions—part I: mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).
[Crossref]

N. Takaki and J. Rolland, “Mathematical properties of describing freeform optical surfaces with orthogonal bases,” Proc. SPIE 10590, 105900U (2017).
[Crossref]

2016 (1)

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

2015 (1)

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

2013 (1)

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Test. 2, 97–109 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (2)

2008 (1)

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
[Crossref]

2007 (1)

G. Forbes, “Shape specification for axially symmetric optical surfaces,” Optics Express 15, 5218–5226 (2007).
[Crossref]

2005 (1)

2004 (1)

M. Bray, “Orthogonal polynomials: a set for square areas,” Proc. SPIE 5252, 314–322 (2004).
[Crossref]

1994 (1)

1981 (1)

1934 (1)

F. Zernike, “Beugungstheorie des Schneidenver-Fahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Bauer, A.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Beier, M.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Benítez, P.

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Bray, M.

M. Bray, “Orthogonal polynomials: a set for square areas,” Proc. SPIE 5252, 314–322 (2004).
[Crossref]

Broemel, A.

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions—part I: mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).
[Crossref]

Bromel, A.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Chen, L.

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

Chow, W. W.

Cohn, B.

L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. patent4,826,287 (2May1989).

Cook, L.

L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. patent4,826,287 (2May1989).

Davies, M.

M. Davies, Mechanical Engineering and Engineering Science, The University of North Carolina at Charlotte (personal communication, 2018).

Dennis, J. E.

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics (Prentice-Hall, Inc., 1983).

Forbes, G.

Forbes, G. W.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Test. 2, 97–109 (2013).
[Crossref]

G. W. Forbes, “Robust and fast computation for the polynomials of optics,” Opt. Express 18, 13851–13862 (2010).
[Crossref]

Fuerschbach, K.

Gao, Z.

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

Grabovickic, D.

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Gross, H.

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions—part I: mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).
[Crossref]

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Gu, P.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
[Crossref]

Hartung, J.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Li, X.

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

Lippman, U.

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions—part I: mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).
[Crossref]

Liu, J.

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Liu, X.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
[Crossref]

Mahajan, V.

Mahajan, V. N.

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 498–546.

Menke, C.

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Test. 2, 97–109 (2013).
[Crossref]

Minano, J.

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Nakano, T.

Narasimhan, B.

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Nikolic, M.

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Ochse, D.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

D. Ochse, K. Uhlendorf, and L. Reichmann, Describing Freeform Surfaces with Orthogonal Functions (Imaging and Applied Optics, 2015).

Oleszko, M.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Perron, G.

L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. patent4,826,287 (2May1989).

Reichmann, L.

D. Ochse, K. Uhlendorf, and L. Reichmann, Describing Freeform Surfaces with Orthogonal Functions (Imaging and Applied Optics, 2015).

Rolland, J.

N. Takaki and J. Rolland, “Mathematical properties of describing freeform optical surfaces with orthogonal bases,” Proc. SPIE 10590, 105900U (2017).
[Crossref]

Rolland, J. P.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing phi-type polynomial surfaces,” Opt. Express 19, 21919 (2011).
[Crossref]

Schiesser, E. M.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Schnabel, R. B.

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics (Prentice-Hall, Inc., 1983).

Steinkopf, R.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Sun, X.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
[Crossref]

Swantner, W.

Takaki, N.

N. Takaki and J. Rolland, “Mathematical properties of describing freeform optical surfaces with orthogonal bases,” Proc. SPIE 10590, 105900U (2017).
[Crossref]

Tamagawa, Y.

Thompson, K. P.

Uhlendorf, K.

D. Ochse, K. Uhlendorf, and L. Reichmann, Describing Freeform Surfaces with Orthogonal Functions (Imaging and Applied Optics, 2015).

Ye, J.

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

Yuan, Q.

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

Zellers, B.

L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. patent4,826,287 (2May1989).

Zernike, F.

F. Zernike, “Beugungstheorie des Schneidenver-Fahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Zheng, Z.

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
[Crossref]

Zhong, Y.

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Adv. Opt. Technol. (1)

A. Broemel, U. Lippman, and H. Gross, “Freeform surface descriptions—part I: mathematical representations,” Adv. Opt. Technol. 6, 327–336 (2017).
[Crossref]

Adv. Opt. Test. (1)

C. Menke and G. W. Forbes, “Optical design with orthogonal representations of rotationally symmetric and freeform aspheres,” Adv. Opt. Test. 2, 97–109 (2013).
[Crossref]

Appl. Opt. (2)

Displays (1)

Z. Zheng, X. Sun, X. Liu, and P. Gu, “Design of reflective projection lens with Zernike polynomials surfaces,” Displays 29, 412–417 (2008).
[Crossref]

J. Opt. Soc. Am. (1)

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for freeform optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Opt. Eng. (2)

J. Ye, L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56, 110901 (2017).
[Crossref]

M. Nikolic, P. Benítez, B. Narasimhan, D. Grabovickic, J. Liu, and J. Minano, “Optical design through optimization for rectangular apertures using freeform orthogonal polynomials: a case study,” Opt. Eng. 55, 071204 (2016).
[Crossref]

Opt. Express (4)

Optics Express (1)

G. Forbes, “Shape specification for axially symmetric optical surfaces,” Optics Express 15, 5218–5226 (2007).
[Crossref]

Physica (1)

F. Zernike, “Beugungstheorie des Schneidenver-Fahrens und Seiner Verbesserten Form, der Phasenkontrastmethode,” Physica 1, 689–704 (1934).
[Crossref]

Proc. SPIE (3)

M. Bray, “Orthogonal polynomials: a set for square areas,” Proc. SPIE 5252, 314–322 (2004).
[Crossref]

N. Takaki and J. Rolland, “Mathematical properties of describing freeform optical surfaces with orthogonal bases,” Proc. SPIE 10590, 105900U (2017).
[Crossref]

H. Gross, A. Bromel, M. Beier, R. Steinkopf, J. Hartung, Y. Zhong, M. Oleszko, and D. Ochse, “Overview on surface representations for freeform surfaces,” Proc. SPIE 9626, 96260U (2015).
[Crossref]

Other (10)

E. W. Weisstein, “Jacobi polynomials,” (from MathWorld—A Wolfram Web Resource), 2018, http://mathworld.wolfram.com/JacobiPolynomial.html .

CODE V, “Zernike polynomials,” in CODE V Lens System Setup Reference Manual (2015), Appendix A.

M. Davies, Mechanical Engineering and Engineering Science, The University of North Carolina at Charlotte (personal communication, 2018).

D. Ochse, K. Uhlendorf, and L. Reichmann, Describing Freeform Surfaces with Orthogonal Functions (Imaging and Applied Optics, 2015).

International Organization for Standardization, ISO 10110-19 Optics and Photonics—Preparation of Drawings for Optical Elements and Systems—Part 19 (ISO, 2015).

E. W. Weisstein, “Hessian” (from MathWorld—A Wolfram Web Resource), 2018, http://mathworld.wolfram.com/Hessian.html .

E. W. Weisstein, “Second derivative test” (from MathWorld—A Wolfram Web Resource), 2018, http://mathworld.wolfram.com/SecondDerivativeTest.html .

J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Series in Computational Mathematics (Prentice-Hall, Inc., 1983).

V. N. Mahajan, “Zernike polynomials and wavefront fitting,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 498–546.

L. Cook, G. Perron, B. Zellers, and B. Cohn, “Display system having coma-control plate in relay lens,” U.S. patent4,826,287 (2May1989).

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Figures (2)

Fig. 1.
Fig. 1. (left) Physically tilting by angle α and (right) introducing Zernike tilt change the local tilt at the origin.
Fig. 2.
Fig. 2. Section of an off-axis parabola; Table 2 includes three 2D-Q descriptions of this off-axis parabola section.

Tables (2)

Tables Icon

Table 1. Four Distinct Sets of Zernike Coefficients Describing a Simple Parabolaa

Tables Icon

Table 2. Three Different 2D-Q Descriptions of the Off-Axis Parabola Section from Fig. 2a

Equations (16)

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

P n P n = 0 when    n n .
f g = π π 0 1 f ( u , θ ) g ( u , θ ) W ( u ) u d u d θ / π π 0 1 W ( u ) u d u d θ ,
f g = 1 π π π 0 1 f ( u , θ ) g ( u , θ ) u d u d θ .
f · g = 1 π 2 π π 0 1 f ( u , θ ) · g ( u , θ ) u d u d θ u 2 ( 1 u 2 ) .
z = f ( ρ , θ ) = c ρ 2 1 + 1 c 2 ρ 2 + 1 1 c 2 ρ 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 ) } .
z = f ( ρ , θ ) = c ρ 2 1 + 1 c 2 ρ 2 + n = 0 N m = n n C n m Z n m ( u , θ ) .
Z n m = u | m | P ( n | m | ) / 2 ( 0 , | m | ) ( 2 u 2 1 ) { cos ( m θ ) if    m 0 sin ( m θ ) if    m < 0 } ,
z = f ( ρ , θ ) = c ρ 2 1 + 1 c 2 ρ 2 + { n ˜ = 0 N ˜ C ˜ n ˜ 0 Z ˜ n ˜ 0 ( u 2 ) + m ˜ = 1 M ˜ u m ˜ n ˜ = 0 N ˜ [ C ˜ n ˜ m ˜ cos ( m ˜ θ ) + C ˜ n ˜ m ˜ sin ( m ˜ θ ) ] Z ˜ n ˜ m ˜ ( u 2 ) } .
2 ρ 2 ( n = 0 N C n 0 Z n 0 ( ρ / ρ max , θ ) ) = 1 ρ max 2 ( n = 0 N ( 1 ) ( n 2 1 ) n ( n 2 + 1 ) C n 0 ) = 0 when    ρ = 0 .
n = 0 N C n 0 Z n 0 ( ρ / ρ max , θ ) = n = 0 N C n 0 = 0 when    ρ = ρ max .
π π δ ( u , θ ) d θ = 0 when    u = 1 .
z = f ( ρ , θ ) = 0 when    ρ = 0 .
n = 0 N m = n n C n m Z n m ( 0 , θ ) = n = 0 N ( 1 ) ( n 2 ) C n 0 = 0 .
a 0 1 = b 0 1 = 0 for the    2 D - Qs , C 1 1 = C 1 1 = 0 for the    Zernikes .
n = 0 N a n 1 Q n 1 ( 0 ) = n = 0 N b n 1 Q n 1 ( 0 ) = 0
n = 0 N ( 1 ) ( n 1 2 ) ( n + 1 2 ) C n 1 = n = 0 N ( 1 ) ( n 1 2 ) ( n + 1 2 ) C n 1 = 0

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