Abstract

From the literature the calculation of power and astigmatism of a local wavefront after refraction at a given surface is known from the vergence and Coddington equations. For higher-order aberrations (HOAs) equivalent analytical equations do not exist. Since HOAs play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the “generalized Coddington equation” to the case of HOA (e.g., coma and spherical aberration). This is done by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the local HOA of an outgoing wavefront directly from the aberrations of the incoming wavefront and the refractive surface.

© 2010 Optical Society of America

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References

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    [CrossRef]
  5. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090-1100 (2009).
    [CrossRef]
  6. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26, 1503-1517 (2009).
    [CrossRef]
  7. R. Krueger, R. Applegate, and S. MacRae, Wavefront Customized Visual Correction (Slack, 2004).
  8. J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006).
    [CrossRef]
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    [CrossRef]
  10. R. Applegate, “Glenn Fry Award Lecture 2002: Wavefront sensing, ideal corrections, and visual Performance,” Optom. Vision Sci. 81, 137-177 (2004).
    [CrossRef]
  11. R. Blendowske, “Wieso funktionieren Gleitsichtgläser? Über Aberrationen in der Progressionszone,” Deutsche Optikerzeitung 2, 60-64 (2007).
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  15. Rodenstock, “Method for computing a progressive spectacle lens and methods for manufacturing a spectacle lens of this kind,” U.S. patent 6,832,834 B2, December 21, 2004.
  16. Rodenstock, “Method for calculating an individual progressive lens,” U.S. patent application 2007/0132945 A1, June 14, 2007.
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    [CrossRef]
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
  23. W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. R. Dorsch, W. Haimerl, and G. Esser, “Accurate computation of mean power and astigmatism by means of Zernike polynomials,” J. Opt. Soc. Am. A 15, 1686-1688 (1998).
    [CrossRef]
  27. E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vision Sci. 82, 923-932 (2005).
    [CrossRef]
  28. W. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vision Sci. 74, 349-366 (1997).
    [CrossRef]
  29. W. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vision Sci. 84, 1060-1063 (2007).
    [CrossRef]
  30. L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
    [CrossRef]
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    [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2009

2008

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

2007

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vision Sci. 84, 72-78 (2007).
[CrossRef]

R. Blendowske, “Wieso funktionieren Gleitsichtgläser? Über Aberrationen in der Progressionszone,” Deutsche Optikerzeitung 2, 60-64 (2007).

R. Blendowske, “Brillengläser und die Korrektion der Abbildungsfehler höherer Ordnung,” Deutsche Optikerzeitung 6, 18-25 (2007).

W. Wesemann, “Korrektion der Aberrationen höherer Ordnung des Auges mit Brillengläsern--Möglichkeiten und Probleme,” Deutsche Optikerzeitung 9, 44-49 (2007).

W. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vision Sci. 84, 1060-1063 (2007).
[CrossRef]

2006

2005

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vision Sci. 82, 923-932 (2005).
[CrossRef]

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005).
[CrossRef]

2004

R. Applegate, “Glenn Fry Award Lecture 2002: Wavefront sensing, ideal corrections, and visual Performance,” Optom. Vision Sci. 81, 137-177 (2004).
[CrossRef]

2001

1998

1997

W. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vision Sci. 74, 349-366 (1997).
[CrossRef]

L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

1996

1982

1981

Acosta, E.

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vision Sci. 84, 72-78 (2007).
[CrossRef]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vision Sci. 82, 923-932 (2005).
[CrossRef]

Altheimer, H.

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

Applegate, R.

R. Applegate, “Glenn Fry Award Lecture 2002: Wavefront sensing, ideal corrections, and visual Performance,” Optom. Vision Sci. 81, 137-177 (2004).
[CrossRef]

R. Krueger, R. Applegate, and S. MacRae, Wavefront Customized Visual Correction (Slack, 2004).

Awwal, A.

J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006).
[CrossRef]

Becken, W.

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

Blendowske, R.

R. Blendowske, “Wieso funktionieren Gleitsichtgläser? Über Aberrationen in der Progressionszone,” Deutsche Optikerzeitung 2, 60-64 (2007).

R. Blendowske, “Brillengläser und die Korrektion der Abbildungsfehler höherer Ordnung,” Deutsche Optikerzeitung 6, 18-25 (2007).

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vision Sci. 84, 72-78 (2007).
[CrossRef]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vision Sci. 82, 923-932 (2005).
[CrossRef]

Born, M.

M. Born and E. Wolf, “Foundations of geometrical optics, geometrical theory of optical imaging and geometrical theory of optical aberrations,” in Principles of Optics (Pergamon, 1980), pp. 109-232.

Burkhard, D.

Cakmakci, O.

Campbell, C.

Cox, I.

Dillon, K.

Dorsch, R.

Esser, G.

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

R. Dorsch, W. Haimerl, and G. Esser, “Accurate computation of mean power and astigmatism by means of Zernike polynomials,” J. Opt. Soc. Am. A 15, 1686-1688 (1998).
[CrossRef]

G. Esser, Derivation of the Imaging Equations for the Calculation of the Higher Order Aberrations of a Local Wavefront after Refraction (Hieronymus, 2008).
[PubMed]

Golub, M. A.

Guirao, A.

Haimerl, W.

Harris, W.

W. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vision Sci. 84, 1060-1063 (2007).
[CrossRef]

W. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vision Sci. 74, 349-366 (1997).
[CrossRef]

Horner, D.

L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Krueger, R.

R. Krueger, R. Applegate, and S. MacRae, Wavefront Customized Visual Correction (Slack, 2004).

Landgrave, J.

Lin, J.

J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006).
[CrossRef]

MacRae, S.

R. Krueger, R. Applegate, and S. MacRae, Wavefront Customized Visual Correction (Slack, 2004).

Mahajan, V. N.

V. N. Mahajan, “Gaussian optics, optical aberrations and calculation of primary aberrations,” in V.MahajanOptical Imaging and Aberrations Part I: Ray Geometrical Optics (SPIE, 1998), pp. 91-361.

Moya-Cessa, J.

Mueller, W.

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

Porter, J.

Quener, H.

J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006).
[CrossRef]

Rolland, J. P.

Schmid, T.

Seidemann, A.

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

Shannon, R.

R. Shannon, “Geometrical optics,” in The Art and Science of Optical Design (Cambridge Univ. Press, 1997), pp. 25-105.

Shealy, D.

Stavroudis, O.

O. Stavroudis, “Surfaces,” in O.Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-160.

Thibos, L.

L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Thompson, K. P.

Thorn, K.

J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006).
[CrossRef]

Tyson, R. K.

Uttenweiler, D.

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

Wesemann, W.

W. Wesemann, “Korrektion der Aberrationen höherer Ordnung des Auges mit Brillengläsern--Möglichkeiten und Probleme,” Deutsche Optikerzeitung 9, 44-49 (2007).

Wheeler, W.

L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

Williams, D.

Wolf, E.

M. Born and E. Wolf, “Foundations of geometrical optics, geometrical theory of optical imaging and geometrical theory of optical aberrations,” in Principles of Optics (Pergamon, 1980), pp. 109-232.

Appl. Opt.

Deutsche Optikerzeitung

R. Blendowske, “Wieso funktionieren Gleitsichtgläser? Über Aberrationen in der Progressionszone,” Deutsche Optikerzeitung 2, 60-64 (2007).

R. Blendowske, “Brillengläser und die Korrektion der Abbildungsfehler höherer Ordnung,” Deutsche Optikerzeitung 6, 18-25 (2007).

W. Wesemann, “Korrektion der Aberrationen höherer Ordnung des Auges mit Brillengläsern--Möglichkeiten und Probleme,” Deutsche Optikerzeitung 9, 44-49 (2007).

J. Opt. Soc. Am. A

J. Landgrave and J. Moya-Cessa, “Generalized Coddington equations in ophthalmic lens design,” J. Opt. Soc. Am. A 13, 1637-1644 (1996).
[CrossRef]

K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005).
[CrossRef]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090-1100 (2009).
[CrossRef]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26, 1503-1517 (2009).
[CrossRef]

J. Porter, A. Guirao, I. Cox, and D. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793-1803 (2001).
[CrossRef]

C. Campbell, “Generalized Coddington equations found via an operator method,” J. Opt. Soc. Am. A 23, 1691-1698 (2006).
[CrossRef]

M. A. Golub, “Analogy between generalized Coddington equations and thin optical element approximation,” J. Opt. Soc. Am. A 26, 1235-1239 (2009).
[CrossRef]

R. Dorsch, W. Haimerl, and G. Esser, “Accurate computation of mean power and astigmatism by means of Zernike polynomials,” J. Opt. Soc. Am. A 15, 1686-1688 (1998).
[CrossRef]

K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839-1846 (2009).
[CrossRef]

Opt. Lett.

Optom. Vision Sci.

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vision Sci. 84, 72-78 (2007).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the paraxial case,” Optom. Vision Sci. 85, 581-592 (2008).
[CrossRef]

W. Becken, H. Altheimer, G. Esser, W. Mueller, and D. Uttenweiler, “Wavefront method for computing the magnification matrix of optical systems: near objects in the general case of strongly oblique incidence,” Optom. Vision Sci. 85, 593-604 (2008).
[CrossRef]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vision Sci. 82, 923-932 (2005).
[CrossRef]

W. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vision Sci. 74, 349-366 (1997).
[CrossRef]

W. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vision Sci. 84, 1060-1063 (2007).
[CrossRef]

L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of Fourier analysis to the description and statistical analysis of refractive error,” Optom. Vision Sci. 74, 367-375 (1997).
[CrossRef]

R. Applegate, “Glenn Fry Award Lecture 2002: Wavefront sensing, ideal corrections, and visual Performance,” Optom. Vision Sci. 81, 137-177 (2004).
[CrossRef]

Z. Med. Phys.

W. Becken, A. Seidemann, H. Altheimer, G. Esser, and D. Uttenweiler, “Spectacle lenses in sports: optimization of the imaging properties based on physiological aspects,” Z. Med. Phys. 17, 56-66 (2007).
[PubMed]

Other

O. Stavroudis, “Surfaces,” in O.Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-160.

Rodenstock, “Spectacle lens with small higher order aberrations,” U.S. patent 7,063,421 B2, June 20, 2006.

Rodenstock, “Method for computing a progressive spectacle lens and methods for manufacturing a spectacle lens of this kind,” U.S. patent 6,832,834 B2, December 21, 2004.

Rodenstock, “Method for calculating an individual progressive lens,” U.S. patent application 2007/0132945 A1, June 14, 2007.

M. Born and E. Wolf, “Foundations of geometrical optics, geometrical theory of optical imaging and geometrical theory of optical aberrations,” in Principles of Optics (Pergamon, 1980), pp. 109-232.

V. N. Mahajan, “Gaussian optics, optical aberrations and calculation of primary aberrations,” in V.MahajanOptical Imaging and Aberrations Part I: Ray Geometrical Optics (SPIE, 1998), pp. 91-361.

R. Shannon, “Geometrical optics,” in The Art and Science of Optical Design (Cambridge Univ. Press, 1997), pp. 25-105.

R. Krueger, R. Applegate, and S. MacRae, Wavefront Customized Visual Correction (Slack, 2004).

J. Porter, H. Quener, J. Lin, K. Thorn, and A. Awwal, Adaptive Optics for Vision Science (Wiley, 2006).
[CrossRef]

G. Esser, Derivation of the Imaging Equations for the Calculation of the Higher Order Aberrations of a Local Wavefront after Refraction (Hieronymus, 2008).
[PubMed]

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Figures (5)

Fig. 1
Fig. 1

Local coordinate systems of the refractive surface of the incoming wavefront and the outgoing wavefront. (a) True situation that the origins of all coordinate systems coincide. (b) Fictitious situation of separated origins for a better understanding of the nomenclature. The surface normal vectors along the neighboring ray are also drawn, referred to as n ¯ In , n ¯ S , n ¯ Out in their preferred local coordinate systems ( x ¯ , y ¯ , z ¯ ) coincide. (c) Meaning of the vector sum in Eq. (25).

Fig. 2
Fig. 2

Orthogonal incidence of a spherical wavefront with vergence S = n s onto a spherical surface with surface power S ¯ .

Fig. 3
Fig. 3

Ray-tracing plots for example A generated by ZEMAX. (a) Spherical surface with radius r = 1 a ¯ S , 2 . (b) Parabolic surface with local curvature a ¯ S , 2 . (c) Strongly reduced aberrations due to aspherical surface of sixth order with coefficients a ¯ S , 2 , a ¯ S , 4 , and a ¯ S , 6 . The vertical lines in the middle of the drawings are construction lines of ZEMAX and have no relevance in our context.

Fig. 4
Fig. 4

Ray-tracing plot for example B generated by ZEMAX. A spherical wavefront is refracted by a spherical surface under oblique incidence, giving rise to coma in the outgoing wavefront. The box drawn around the refractive object consists of construction lines of ZEMAX and has no relevance in our context.

Fig. 5
Fig. 5

Relationship between the sagitta w ( y ) of a wavefront and its OPD given by the function τ w ( y ¯ ) .

Tables (2)

Tables Icon

Table 1 Local Aberrations of the Outgoing Wavefront in Example B a

Tables Icon

Table 2 Zernike Coefficients of the Outgoing Wavefront in Example B a

Equations (164)

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

v = R ( ϵ ) v ¯ , v = R ( ϵ ) v ¯ ,
R ( ϵ ) = ( 1 0 0 0 cos ϵ sin ϵ 0 sin ϵ cos ϵ ) .
w ( x , y ) = k = 0 m = k k c k m Z k m ( ρ , ϑ ) , ( m k ) even ,
w ( x , y ) = k = 0 m = 0 k a m , k m m ! ( k m ) ! x m y k m ,
a m , k m = | k x m y k m w ( x , y ) | x = 0 , y = 0 = w ( m , k m ) ( 0 , 0 ) .
w In ( x , y ) = k = 0 m = 0 k a In , m , k m m ! ( k m ) ! x m y k m ,
w Out ( x , y ) = k = 0 m = 0 k a Out , m , k m m ! ( k m ) ! x m y k m ,
w ¯ S ( x ¯ , y ¯ ) = k = 0 m = 0 k a ¯ m , k m m ! ( k m ) ! x ¯ m y ¯ k m
S = S + S ¯ ,
n ( w In ( 2 , 0 ) w In ( 1 , 1 ) w In ( 1 , 1 ) w In ( 2 , 0 ) ) ,
s = ( S x x S x y S y y ) = n ( w In ( 2 , 0 ) w In ( 1 , 1 ) w In ( 0 , 2 ) ) , s = ( S x x S x y S y y ) = n ( w Out ( 2 , 0 ) w Out ( 1 , 1 ) w Out ( 0 , 2 ) ) ,
s ¯ = ( S ¯ x x S ¯ x y S ¯ y y ) = ( n n ) ( w ¯ S ( 2 , 0 ) w ¯ S ( 1 , 1 ) w ¯ S ( 0 , 2 ) ) .
S x x = ( Sph + Cyl 2 ) Cyl 2 cos 2 α ,
S x y = Cyl 2 sin 2 α ,
S y y = ( Sph + Cyl 2 ) + Cyl 2 cos 2 α ,
C s = C s + ν s ¯ ,
C = ( 1 0 0 0 cos ϵ 0 0 0 cos 2 ϵ ) , C = ( 1 0 0 0 cos ϵ 0 0 0 cos 2 ϵ )
ν = n cos ϵ n cos ϵ n n .
e k = ( E x x x E x x y E y y y ) n ( w In ( k , 0 ) w In ( k 1 , 1 ) w In ( 0 , k ) ) ,
e k = ( E x x x E x x y E y y y ) n ( w Out ( k , 0 ) w Out ( k 1 , 1 ) w Out ( 0 , k ) ) ,
e ¯ k = ( E ¯ x x x E ¯ x x y E ¯ y y y ) ( n n ) ( w ¯ S ( k , 0 ) w ¯ S ( k 1 , 1 ) w ¯ S ( 0 , k ) ) ,
C k = ( 1 0 0 0 cos ϵ 0 cos k ϵ ) ,
C k = ( 1 0 0 0 cos ϵ 0 cos k ϵ ) .
R ( ϵ ) = ( cos ϵ sin ϵ sin ϵ cos ϵ ) .
w In ( y ) = ( y w In ( y ) ) ,
w In ( y ) = k = 0 a In , k k ! y k ,
w Out ( y ) = ( y w Out ( y ) ) , w ¯ S ( y ¯ ) = ( y ¯ w ¯ S ( y ¯ ) ) ,
w Out ( y ) = k = 0 a Out , k k ! y k , w ¯ S ( y ¯ ) = k = 0 a ¯ S , k k ! y ¯ k .
a In , k = | k y k w In ( y ) | y = 0 = w In ( k ) ( 0 ) .
n ( v ) 1 1 + ν 2 ( v 1 ) .
n ( 0 ) ( 0 1 ) , n ( 1 ) ( 0 ) ( 1 0 ) , n ( 2 ) ( 0 ) ( 0 1 ) ,
n ( 3 ) ( 0 ) ( 3 0 ) , n ( 4 ) ( 0 ) ( 0 9 ) , etc.
| y n In ( y ) | y = 0 n In ( 1 ) ( 0 ) = n ( 1 ) ( 0 ) w In ( 2 ) ( 0 ) = ( 1 0 ) w In ( 2 ) ( 0 ) ,
| y n Out ( y ) | y = 0 n Out ( 1 ) ( 0 ) = n ( 1 ) ( 0 ) w Out ( 2 ) ( 0 ) = ( 1 0 ) w Out ( 2 ) ( 0 ) ,
| y ¯ n ¯ S ( y ¯ ) | y ¯ = 0 n ¯ S ( 1 ) ( 0 ) = n ( 1 ) ( 0 ) w ¯ S ( 2 ) ( 0 ) = ( 1 0 ) w ¯ S ( 2 ) ( 0 ) ,
( y In w In ( y In ) ) τ n n In = R ( ϵ ) ( y ¯ S w ¯ S ( y ¯ S ) ) ,
( y Out w Out ( y Out ) ) τ n n Out = R ( ϵ ) ( y ¯ S w ¯ S ( y ¯ S ) ) .
p ( y ¯ S ) = ( y In ( y ¯ S ) y Out ( y ¯ S ) τ ( y ¯ S ) w ¯ S ( y ¯ S ) )
f ( p , y ¯ S ) = ( y In τ n n y ( w In ( 1 ) ( y In ) ) ( y ¯ S cos ϵ w ¯ S sin ϵ ) w In ( y In ) τ n n z ( w In ( 1 ) ( y In ) ) ( y ¯ S sin ϵ + w ¯ S cos ϵ ) y Out τ n n y ( w Out ( 1 ) ( y Out ) ) ( y ¯ S cos ϵ w ¯ S sin ϵ ) w Out ( y Out ) τ n n z ( w Out ( 1 ) ( y Out ) ) ( y ¯ S sin ϵ + w ¯ S cos ϵ ) ) ,
f ( p ( y ¯ S ) , y ¯ S ) = 0 ,
j = 1 4 f i p j p j ( 1 ) ( y ¯ S ) + f i y ¯ S = 0 , i = 1 , , 4 ,
A ( f 1 y In f 1 y Out f 1 τ f 1 w ¯ S f 2 y In f 2 y Out f 2 τ f 2 w ¯ S f 3 y In f 3 y Out f 3 τ f 3 w ¯ S f 4 y In f 4 y Out f 4 τ f 4 w ¯ S ) = ( 1 τ n n In , y ( 1 ) w In ( 2 ) 0 1 n n In , y σ w In ( 1 ) τ n n In , z ( 1 ) w In ( 2 ) 0 1 n n In , z χ 0 1 τ n n Out , y ( 1 ) w Out ( 2 ) 1 n n Out , y σ 0 w Out ( 1 ) τ n n Out , z ( 1 ) w Out ( 2 ) 1 n n Out , z χ ) .
b f y ¯ S = ( χ σ χ σ ) ,
A ( p ( y ¯ S ) ) p ( 1 ) ( y ¯ S ) = b .
p ( 1 ) ( y ¯ S ) = A ( p ( y ¯ S ) ) 1 b .
p ( 1 ) ( 0 ) = A 1 b ,
p ( 2 ) ( 0 ) = ( A 1 ) ( 1 ) b ,
p ( k ) ( 0 ) = ( A 1 ) ( k 1 ) b ,
A 1 = A ( p ( 0 ) ) 1 = A ( 0 ) 1 ,
( A 1 ) ( 1 ) = | d d y ¯ S A ( p ( y ¯ S ) ) 1 | y ¯ S = 0 , ,
( A 1 ) ( k 1 ) = | d k 1 d y ¯ S k 1 A ( p ( y ¯ S ) ) 1 | y ¯ S = 0
p ( 2 ) ( 0 ) = | i = 1 4 ( p i A ( p ) 1 ) p i ( 1 ) | y ¯ S = 0 b ,
A p ( 1 ) ( 0 ) = b , ( a )
A ( 1 ) p ( 1 ) ( 0 ) + A p ( 2 ) ( 0 ) = 0 , ( b )
A ( 2 ) p ( 1 ) ( 0 ) + 2 A ( 1 ) p ( 2 ) ( 0 ) + A p ( 3 ) ( 0 ) = 0 , ( c )
j = 1 k ( k 1 j 1 ) A ( k j ) p ( j ) ( 0 ) = 0 , k 2 ( d ) ,
A = A ( p ( 0 ) ) = A ( 0 ) ,
A ( 1 ) = | d d y ¯ S A ( p ( y ¯ S ) ) | y ¯ S = 0 , ,
A ( k j ) = | d k j d y ¯ S k j A ( p ( y ¯ S ) ) | y ¯ S = 0
( f g ) ( p ) = j = 0 p ( p j ) f ( p j ) g ( j ) .
p ( 1 ) ( 0 ) = A 1 b , k = 1 ,
p ( k ) ( 0 ) = A 1 j = 1 k 1 ( k 1 j 1 ) A ( k j ) p ( j ) ( 0 ) , k 2 .
A ( 0 ) = ( 1 0 0 σ 0 0 1 n χ 0 1 0 σ 0 0 1 n χ )
A ( 0 ) 1 = ( 1 n σ η 0 n σ η 0 n σ η 1 n σ η 0 n n χ η 0 n n χ η 0 n η 0 n η ) with η = n χ n χ .
p ( 1 ) ( 0 ) = ( χ χ n σ 0 ) .
η w ¯ S ( 2 ) = χ 2 n w Out ( 2 ) χ 2 n w In ( 2 ) ,
η w ¯ S ( 3 ) = χ 3 n w Out ( 3 ) χ 3 n w In ( 3 ) + R 3 ,
η w ¯ S ( 4 ) = χ 4 n w Out ( 4 ) χ 4 n w In ( 4 ) + R 4 ,
η w ¯ S ( k ) = χ k n w Out ( k ) χ k n w In ( k ) + R k ,
R 3 = 3 n σ χ χ η ( n w Out ( 2 ) n w In ( 2 ) ) ( χ w Out ( 2 ) χ w In ( 2 ) ) ,
R 4 = ( α w Out ( 2 ) + β w In ( 2 ) ) w Out ( 3 ) + ( β w Out ( 2 ) + α w In ( 2 ) ) w In ( 3 ) + γ ( w Out ( 2 ) ) 3 + δ ( w Out ( 2 ) ) 2 w In ( 2 ) + δ w Out ( 2 ) ( w In ( 2 ) ) 2 + γ ( w In ( 2 ) ) 3 ,
α = 2 n σ χ 3 η ( n χ 6 n χ ) , β = 2 n σ χ χ 2 η ( 2 n χ + 3 n χ ) ,
γ = 3 n χ 2 η 2 ( 2 n χ 2 χ η σ 2 ( n 2 χ 2 + 4 n 2 χ 2 ) ) ,
δ = 3 n χ n η 2 ( ( 2 n χ + n χ ) ( n 2 χ 2 + 2 n n χ χ + 2 n 2 χ 2 ) σ 2 2 η ( n χ χ ) 2 ) ,
ν E ¯ k = E k cos k ϵ E k cos k ϵ + R k ,
p ( k ) = A 1 A ( k 1 ) p ( 1 ) = ( A 1 A ( k 1 ) A 1 ) b k 2 .
A ( m ) = ( m χ m 1 σ w In ( m + 1 ) 0 χ m n w In ( m + 1 ) 0 χ m w In ( m + 1 ) 0 0 0 0 m χ m 1 σ w Out ( m + 1 ) χ m n w In ( m + 1 ) 0 0 χ m w In ( m + 1 ) 0 0 )
η s ¯ ( k ) ( 0 ) = χ k n w Out ( k ) ( 0 ) χ k n w In ( k ) ( 0 )
ν E ¯ k = E k cos k ϵ E k cos k ϵ ,
w In ( x , y ) = ( x y w In ( x , y ) ) ,
n ( u , v ) 1 1 + u 2 + v 2 ( u v 1 ) ,
w ( 1 , 0 ) × w ( 0 , 1 ) | w ( 1 , 0 ) × w ( 0 , 1 ) | = 1 1 + w ( 1 , 0 ) 2 + w ( 0 , 1 ) 2 ( w ( 1 , 0 ) w ( 0 , 1 ) 1 ) = n ( w ( 1 , 0 ) , w ( 0 , 1 ) ) = n ( w ) .
p ( x ¯ S , y ¯ S ) = ( x In ( x ¯ S , y ¯ S ) y In ( x ¯ S , y ¯ S ) x Out ( x ¯ S , y ¯ S ) y Out ( x ¯ S , y ¯ S ) τ ( x ¯ S , y ¯ S ) s ¯ ( x ¯ S , y ¯ S ) ) ,
f ( p ( x ¯ S , y ¯ S ) , x ¯ S , y ¯ S ) = 0 ,
A ( p ( x ¯ S , y ¯ S ) ) p ( 1 , 0 ) ( x ¯ S , y ¯ S ) = b x ,
A ( p ( x ¯ S , y ¯ S ) ) p ( 0 , 1 ) ( x ¯ S , y ¯ S ) = b y ,
b x = f x ¯ S = ( 1 0 0 1 0 0 ) T ,
b y = f y ¯ S = ( 0 χ σ 0 χ σ ) T .
| A ( p ( x ¯ S , y ¯ S ) ) = ( A In 0 0 A Out | A τ A ¯ S ) ,
A In = ( 1 τ n ( n In , x ( 0 , 1 ) w In ( 1 , 1 ) + n In , x ( 1 , 0 ) w In ( 2 , 0 ) ) τ n ( n In , z ( 0 , 1 ) w In ( 0 , 2 ) + n In , z ( 1 , 0 ) w In ( 1 , 1 ) ) τ n ( n In , y ( 0 , 1 ) w In ( 1 , 1 ) + n In , y ( 1 , 0 ) w In ( 2 , 0 ) ) 1 τ n ( n In , y ( 0 , 1 ) w In ( 0 , 2 ) + n In , y ( 1 , 0 ) w In ( 1 , 1 ) ) w In ( 1 , 0 ) τ n ( n In , z ( 0 , 1 ) w In ( 1 , 1 ) + n In , z ( 1 , 0 ) w In ( 2 , 0 ) ) w In ( 0 , 1 ) τ n ( n In , z ( 0 , 1 ) w In ( 0 , 2 ) + n In , z ( 1 , 0 ) w In ( 1 , 1 ) ) ) ,
A τ = ( n In , x n n In , y n n In , z n n Out , x n n Out , y n n Out , z n ) , A ¯ S = ( 0 σ χ 0 σ χ ) .
p ( 1 , 0 ) ( 0 , 0 ) = A 1 b x ,
p ( 0 , 1 ) ( 0 , 0 ) = A 1 b y ,
p ( 2 , 0 ) ( 0 , 0 ) = | ( A 1 ) ( 1 , 0 ) | b x ,
p ( 1 , 1 ) ( 0 , 0 ) = ( A 1 ) ( 0 , 1 ) b x = | ( A 1 ) ( 1 , 0 ) | b y ,
p ( 0 , 2 ) ( 0 , 0 ) = ( A 1 ) ( 0 , 1 ) b y ,
p ( k x , k y ) ( 0 , 0 ) = { ( A 1 ) ( k x 1 , 0 ) b x , k x 0 , k y = 0 , ( A 1 ) ( k x 1 , k y ) b x = ( A 1 ) ( k x , k y 1 ) b y , k x 0 , k y 0 , ( A 1 ) ( 0 , k y 1 ) b y , k x = 0 , k y 0 , }
A 1 = A ( p ( 0 , 0 ) ) 1 = A ( 0 ) 1 , and ( A 1 ) ( 1 , 0 ) = | d d x ¯ S A ( p ( x ¯ S , y ¯ S ) ) 1 | x ¯ S = 0 , y ¯ S = 0 , ( A 1 ) ( k x , k y ) = | d k x d x ¯ S k x d k y d x ¯ S k y A ( p ( x ¯ S , y ¯ S ) ) 1 | x ¯ S = 0 , y ¯ S = 0 ,
A ( 0 ) = ( 1 0 0 0 0 0 0 1 0 0 0 σ 0 0 0 0 1 n χ 0 0 1 0 0 0 0 0 0 1 0 σ 0 0 0 0 1 n χ )
A ( 0 ) 1 = ( 1 0 0 0 0 0 0 1 n σ η 0 0 n σ η 0 0 0 1 0 0 0 0 n σ η 0 1 n σ η 0 0 n n χ η 0 0 n n χ η 0 0 n η 0 0 n η ) ,
p ( 1 , 0 ) ( 0 , 0 ) = ( 1 0 1 0 0 0 ) , p ( 0 , 1 ) ( 0 , 0 ) = ( 0 χ 0 χ n σ 0 ) .
η w ¯ S ( k x , k y ) = χ k y n w Out ( k x , k y ) χ k y n w In ( k x , k y ) + R k x , k y .
ν e ¯ k = C k e k C k e k + r k ,
p ( k x , 0 ) ( 0 , 0 ) = A 1 j x = 1 k x 1 ( k x 1 j x 1 ) A ( k x j x , 0 ) p ( j x , 0 ) , k x 2 , k y = 0 ,
p ( k x , k y ) ( 0 , 0 ) = A 1 j x 1 , j y 0 j x + j y < k x + k y ( k x 1 j x 1 ) ( k y j y ) A ( k x j x , k y j y ) p ( j x , j y )
= A 1 j x 0 , j y 1 j x + j y < k x + k y ( k x j x ) ( k y 1 j y 1 ) A ( k x j x , k y j y ) p ( j x , j y ) , k x 0 , k y 0 ,
p ( 0 , k y ) ( 0 , 0 ) = A 1 j y = 1 k y 1 ( k y 1 j y 1 ) A ( 0 , k y j y ) p ( 0 , j y ) , k x = 0 , k y 2 ,
p ( k x , 0 ) ( 0 , 0 ) = A 1 A ( k x 1 , 0 ) p ( 1 , 0 ) , k x 2 , k y = 0 ,
p ( k x , k y ) ( 0 , 0 ) = A 1 A ( k x 1 , k y ) p ( 1 , 0 ) = A 1 A ( k x , k y 1 ) p ( 0 , 1 ) , k x 0 , k y 0 ,
p ( 0 , k y ) ( 0 , 0 ) = A 1 A ( 0 , k y 1 ) p ( 0 , 1 ) , k x = 0 , k y 2 .
A ( m x , m y ) = ( | A In ( m x , m y ) 0 0 A Out ( m x , m y ) | A τ ( m x , m y ) A ¯ S ( m x , m y ) )
A In ( m x , m y ) = ( m y χ m y 1 σ w In ( m x + 2 , m y 1 ) m y χ m y 1 σ w In ( m x + 1 , m y ) m y χ m y 1 σ w In ( m x + 1 , m y ) m y χ m y 1 σ w In ( m x , m y + 1 ) χ m y w In ( m x + 1 , m y ) χ m y w In ( m x , m y + 1 ) )
A τ = ( χ m y w In ( m x + 1 , m y ) n χ m y w In ( m x , m y + 1 ) n 0 χ m y w Out ( m x + 1 , m y ) n χ m y w Out ( m x , m y + 1 ) n 0 ) , A ¯ S = 0 ,
η w ¯ S ( k x , k y ) = χ k y n w Out ( k x , k y ) χ k y n w In ( k x , k y )
ν e ¯ k = C k e k C k e k ,
E k cos k ϵ = E k cos k ϵ + ν E ¯ k .
E k cos k ϵ = E k cos k ϵ + ν E ¯ k R k .
S cos 2 ϵ = S cos 2 ϵ + n cos ϵ n cos ϵ n n S ¯ .
E 3 cos 3 ϵ = E 3 cos 3 ϵ + n cos ϵ n cos ϵ n n E ¯ 3 R 3 ,
R 3 = 3 n sin ϵ cos ϵ cos ϵ n cos ϵ n cos ϵ ( n n S n n S ) ( cos ϵ n S cos ϵ n S ) .
C k e k = C k e k + ν e ¯ k r k ,
f ( y ) = r ( 1 1 y 2 r 2 ) ,
f ( y ) = 1 2 r y 2 + 1 8 r 3 y 4 + 1 16 r 5 y 6 + .
a In , 2 = 1 s = S n , a In , 4 = 3 1 s 3 = 3 ( S n ) 3 , a In , 6 = 45 1 s 5 = 45 ( S n ) 5 ,
a Out , 2 = 1 s = S n , a Out , 4 = 3 1 s 3 = 3 ( S n ) 3 , a Out , 6 = 45 1 s 5 = 45 ( S n ) 5 .
s ¯ ( y ) = a ¯ S , 2 2 y 2 + a ¯ S , 4 24 y 4 + a ¯ S , 6 720 y 6 +
a ¯ S , 2 = S S n n .
α = 0 , β = 0 , γ = 6 n n n n , δ = 6 n n n n ,
R 4 = 6 n n n n ( w Out ( 2 ) w In ( 2 ) ) 2 ( w Out ( 2 ) + w In ( 2 ) ) .
a ¯ S , 4 = w ¯ S ( 4 ) = 1 n n ( n w Out ( 4 ) n w In ( 4 ) + R 4 ) = 1 n n ( n a Out , 4 n a In , 4 + 6 n n n n ( a Out , 2 a In , 2 ) 2 ( a Out , 2 + a In , 2 ) ) = 3 ( n n ) 2 ( ( n + n ) S 3 n 2 2 S 2 S n 2 S S 2 n + ( n + n ) S 3 n 2 ) .
a ¯ S , 6 = w ¯ S ( 6 ) = 45 ( n n ) 3 ( ( n + n ) 2 S 5 n 4 + 3 ( n + n ) S 4 S n 3 ( n 3 n ) S 3 S 2 n 2 n + ( n + n ) S 4 n 4 3 ( n + n ) S S 4 n 3 + ( n 3 n ) S 2 S 3 n n 2 ) .
e k OPD = ( E x x x OPD E x x y OPD E y y y OPD ) = ( τ In ( k , 0 ) τ In ( k 1 , 1 ) τ In ( 0 , k ) ) ,
e k OPD = ( E x x x OPD E x x y OPD E y y y OPD ) = ( τ Out ( k , 0 ) τ Out ( k 1 , 1 ) τ Out ( 0 , k ) ) ,
c k m = 1 π r 0 2 pupil Z k m ( x r 0 , y r 0 ) w ( x , y ) d x d y ,
( c 0 0 c 1 1 c 1 1 c 2 2 c 2 0 c 2 2 c 3 3 c 3 1 c k k ) = T ( k ) ( E r 0 E x r 0 E y r 0 2 E x x r 0 2 E x y r 0 2 E y y r 0 3 E x x x r 0 3 E x x y r 0 k E y y y ) = n T ( k ) ( a 00 r 0 a 10 r 0 a 01 r 0 2 a 20 r 0 2 a 11 r 0 2 a 02 r 0 3 a 30 r 0 3 a 21 r 0 k a 0 k ) .
T 1 ( 3 ) = ( 1 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 4 2 0 0 0 0 2 0 0 0 0 0 4 2 0 0 0 0 0 4 3 2 6 0 0 0 0 0 0 0 4 6 0 0 0 0 0 0 0 0 0 0 4 3 2 6 0 0 0 0 0 0 0 0 0 0 12 2 36 2 0 0 0 0 0 0 0 0 0 0 12 2 12 2 0 0 0 0 0 0 12 2 12 2 0 0 0 0 0 0 0 0 0 0 36 2 12 2 ) .
( y w ( y ) ) τ w n n w = ( y ¯ t 0 ) .
τ w ( y ¯ t ( y ) ) = n w ( y ) 1 + w ( 1 ) ( y ) 2 .
y ¯ t ( y ) = y + w ( y ) w ( 1 ) ( y ) .
τ w ( y + w ( y ) w ( 1 ) ( y ) ) = n w ( y ) 1 + w ( 1 ) ( y ) 2 .
τ w ( 0 ) = 0 ,
τ w ( 1 ) ( 0 ) = 0 ,
τ w ( 2 ) ( 0 ) = n w ( 2 ) ( 0 ) ,
3 w ( 2 ) ( 0 ) 2 τ w ( 1 ) ( 0 ) + τ w ( 3 ) ( 0 ) = n w ( 3 ) ( 0 ) ,
10 w ( 2 ) ( 0 ) w ( 3 ) ( 0 ) τ w ( 1 ) ( 0 ) + 12 w ( 2 ) ( 0 ) 2 τ w ( 2 ) ( 0 ) + τ w ( 4 ) ( 0 ) = n ( 6 w ( 2 ) ( 0 ) 3 + w ( 4 ) ( 0 ) ) ,
,
τ w = 0 ,
τ w ( 1 ) = 0 ,
τ w ( 2 ) = n w ( 2 ) ,
τ w ( 3 ) = n w ( 3 ) ,
τ w ( 4 ) = n ( w ( 4 ) 6 w ( 2 ) 3 ) ,
τ w ( 5 ) = n ( w ( 5 ) 40 w ( 2 ) 2 w ( 3 ) ) ,
.
τ w = 0 ,
( τ w ( 1 , 0 ) τ w ( 0 , 2 ) ) = ( 0 0 ) ,
( τ w ( 2 , 0 ) τ w ( 1 , 1 ) τ w ( 0 , 2 ) ) = n ( w ( 2 , 0 ) w ( 1 , 1 ) w ( 0 , 2 ) ) ,
( τ w ( 3 , 0 ) τ w ( 2 , 1 ) τ w ( 1 , 2 ) τ w ( 0 , 3 ) ) = n ( w ( 3 , 0 ) w ( 2 , 1 ) w ( 1 , 2 ) w ( 0 , 3 ) ) ,
( τ w ( 4 , 0 ) τ w ( 3 , 1 ) τ w ( 2 , 2 ) τ w ( 1 , 3 ) τ w ( 0 , 4 ) ) = n ( ( w ( 4 , 0 ) w ( 3 , 1 ) w ( 2 , 2 ) w ( 1 , 3 ) w ( 0 , 4 ) ) ( 6 w ( 2 , 0 ) ( w ( 1 , 1 ) 2 + w ( 2 , 0 ) 2 ) 3 w ( 1 , 1 ) ( w ( 1 , 1 ) 2 + w ( 2 , 0 ) ( w ( 0 , 2 ) + 2 w ( 2 , 0 ) ) ) ( w ( 0 , 2 ) + w ( 2 , 0 ) ) ( 5 w ( 1 , 1 ) 2 + w ( 0 , 2 ) w ( 2 , 0 ) ) 3 w ( 1 , 1 ) ( w ( 1 , 1 ) 2 + w ( 0 , 2 ) ( 2 w ( 0 , 2 ) + w ( 2 , 0 ) ) ) 6 w ( 0 , 2 ) ( w ( 1 , 1 ) 2 + w ( 0 , 2 ) 2 ) ) ) ,
.
r 3 = ( 0 sin ϵ ( n cos ϵ S x x ( n 2 S x x n 2 S ) + n cos ϵ ( n S 2 n 2 ( S 2 x x + S S y y S x x S y y ) ) ) n n 2 ( n cos ϵ n cos ϵ ) 0 3 cos ϵ cos ϵ sin ϵ ( n cos ϵ S n cos ϵ S y y ) ( n 2 S n 2 S y y ) n n 2 ( n cos ϵ n cos ϵ ) ) .

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