Abstract

From the literature the analytical calculation of local power and astigmatism of a wavefront after refraction and propagation is well known; it is, e.g., performed by the Coddington equation for refraction and the classical vertex correction formula for propagation. Recently the authors succeeded in extending the Coddington equation to higher order aberrations (HOA). However, equivalent analytical propagation equations for HOA do not exist. Since HOA play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the propagation equations of power and astigmatism to the case of HOA (e.g., coma and spheri cal aberration). This is achieved by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the aberrations of a propagated wavefront directly from the aberrations of the original wavefront containing both low-order and high-order aberrations.

© 2011 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 109–232.
  2. V. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE, 1998), pp. 91–361.
  3. G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence,” J. Opt. Soc. Am. A 27, 218–237 (2010).
    [CrossRef]
  4. 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).
    [CrossRef] [PubMed]
  5. W. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606–612 (1996).
    [CrossRef] [PubMed]
  6. E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82, 923–32 (2005).
    [CrossRef] [PubMed]
  7. E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72 (2007).
    [CrossRef]
  8. H. Diepes and R. Blendowske, Optik und Technik der Brille (Optische Fachveröffentlichung GmbH, 2002), pp. 477–486.
  9. L. Thibos, “Propagation of astigmatic wavefronts using power vectors,” S. Afr. Optom. 62, 111–113 (2003).
  10. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry; the comatic aberrations,” J. Opt. Soc. Am. A 27, 1490–1504 (2010).
    [CrossRef]
  11. K. P. Thompson, “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]
  12. J. Arasa and J. Alda, “Real ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120027488.
    [CrossRef]
  13. J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120009643.
    [CrossRef]
  14. G. Dai, Wavefront Optics for Vision Correction (SPIE, 2008), pp. 129–255.
    [CrossRef]
  15. K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839–1846 (2009).
    [CrossRef]
  16. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
    [CrossRef]
  17. C. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
    [CrossRef]
  18. G. Dai, “Scaling Zernike expansions coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. 23, 539–543 (2006).
    [CrossRef]
  19. H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. 23, 1960–1968(2006).
    [CrossRef]
  20. S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
    [CrossRef]
  21. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. 24, 569–577 (2007).
    [CrossRef]
  22. A. Guirao, D. Williams, and I. Cox, “Effect of the rotation and translation on the expected benefit of an ideal method to correct the eye’s high-order aberrations,” J. Opt. Soc. Am. A 18, 1003–1015 (2001).
    [CrossRef]
  23. G. Dai, C. Campbell, L. Chen, H. Zhao, and D. Chernyak, “Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials,” Appl. Opt. 48, 477–488 (2009).
    [CrossRef] [PubMed]
  24. W. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vis. Sci. 84, 1060–1063 (2007).
    [CrossRef] [PubMed]
  25. W. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vis. Sci. 74, 349–366 (1997).
    [CrossRef] [PubMed]
  26. L. Thibos, W. Wheeler, and D. Horner, “Power vectors: an application of fourier analysis to the description and statistical analysis of refractive error,” Optom. Vis. Sci. 74, 367–375(1997).
    [CrossRef] [PubMed]
  27. G. M. Dai, “Wavefront expansion basis functions and their relationships,” J. Opt. Soc. Am. A 23, 1657–1666 (2006).
    [CrossRef]
  28. G.-M. Dai, “Wavefront expansion basis functions and their relationships: errata,” J. Opt. Soc. Am. A 23, 2970–2971 (2006).
    [CrossRef]

2010

2009

2007

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

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).
[CrossRef] [PubMed]

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72 (2007).
[CrossRef]

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. 24, 569–577 (2007).
[CrossRef]

2006

G. M. Dai, “Wavefront expansion basis functions and their relationships,” J. Opt. Soc. Am. A 23, 1657–1666 (2006).
[CrossRef]

G.-M. Dai, “Wavefront expansion basis functions and their relationships: errata,” J. Opt. Soc. Am. A 23, 2970–2971 (2006).
[CrossRef]

G. Dai, “Scaling Zernike expansions coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. 23, 539–543 (2006).
[CrossRef]

H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. 23, 1960–1968(2006).
[CrossRef]

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
[CrossRef]

2005

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82, 923–32 (2005).
[CrossRef] [PubMed]

2003

2002

2001

1997

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

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

1996

W. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606–612 (1996).
[CrossRef] [PubMed]

Acosta, E.

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72 (2007).
[CrossRef]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82, 923–32 (2005).
[CrossRef] [PubMed]

Alda, J.

J. Arasa and J. Alda, “Real ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120027488.
[CrossRef]

J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120009643.
[CrossRef]

Altheimer, H.

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).
[CrossRef] [PubMed]

Arasa, J.

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence,” J. Opt. Soc. Am. A 27, 218–237 (2010).
[CrossRef]

J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120009643.
[CrossRef]

J. Arasa and J. Alda, “Real ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120027488.
[CrossRef]

Ares, J.

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
[CrossRef]

Arines, J.

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
[CrossRef]

Bara, S.

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
[CrossRef]

Baumbach, P.

Becken, W.

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence,” J. Opt. Soc. Am. A 27, 218–237 (2010).
[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).
[CrossRef] [PubMed]

Blendowske, R.

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72 (2007).
[CrossRef]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82, 923–32 (2005).
[CrossRef] [PubMed]

H. Diepes and R. Blendowske, Optik und Technik der Brille (Optische Fachveröffentlichung GmbH, 2002), pp. 477–486.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 109–232.

Campbell, C.

Chen, L.

Chernyak, D.

Cox, I.

Dai, G.

G. Dai, C. Campbell, L. Chen, H. Zhao, and D. Chernyak, “Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials,” Appl. Opt. 48, 477–488 (2009).
[CrossRef] [PubMed]

G. Dai, “Scaling Zernike expansions coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. 23, 539–543 (2006).
[CrossRef]

G. Dai, Wavefront Optics for Vision Correction (SPIE, 2008), pp. 129–255.
[CrossRef]

Dai, G. M.

Dai, G.-M.

Diepes, H.

H. Diepes and R. Blendowske, Optik und Technik der Brille (Optische Fachveröffentlichung GmbH, 2002), pp. 477–486.

Dillon, K.

Esser, G.

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence,” J. Opt. Soc. Am. A 27, 218–237 (2010).
[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).
[CrossRef] [PubMed]

Guirao, A.

Han, G.

H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. 23, 1960–1968(2006).
[CrossRef]

Harris, W.

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

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

W. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606–612 (1996).
[CrossRef] [PubMed]

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. Vis. Sci. 74, 367–375(1997).
[CrossRef] [PubMed]

Lundström, L.

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. 24, 569–577 (2007).
[CrossRef]

Luo, L.

H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. 23, 1960–1968(2006).
[CrossRef]

Mahajan, V.

V. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE, 1998), pp. 91–361.

Müller, W.

Prado, P.

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
[CrossRef]

Schwiegerling, J.

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).
[CrossRef] [PubMed]

Shu, H.

H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. 23, 1960–1968(2006).
[CrossRef]

Thibos, L.

L. Thibos, “Propagation of astigmatic wavefronts using power vectors,” S. Afr. Optom. 62, 111–113 (2003).

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

Thompson, K. P.

Unsbo, P.

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. 24, 569–577 (2007).
[CrossRef]

Uttenweiler, D.

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence,” J. Opt. Soc. Am. A 27, 218–237 (2010).
[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).
[CrossRef] [PubMed]

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. Vis. Sci. 74, 367–375(1997).
[CrossRef] [PubMed]

Williams, D.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 109–232.

Zhao, H.

Appl. Opt.

J. Opt. Soc. Am.

G. Dai, “Scaling Zernike expansions coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. 23, 539–543 (2006).
[CrossRef]

H. Shu, L. Luo, and G. Han, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. 23, 1960–1968(2006).
[CrossRef]

S. Bara, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displace pupils,” J. Opt. Soc. Am. 23, 2061–2066 (2006).
[CrossRef]

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. 24, 569–577 (2007).
[CrossRef]

J. Opt. Soc. Am. A

A. Guirao, D. Williams, and I. Cox, “Effect of the rotation and translation on the expected benefit of an ideal method to correct the eye’s high-order aberrations,” J. Opt. Soc. Am. A 18, 1003–1015 (2001).
[CrossRef]

K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry; the comatic aberrations,” J. Opt. Soc. Am. A 27, 1490–1504 (2010).
[CrossRef]

K. P. Thompson, “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]

G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence,” J. Opt. Soc. Am. A 27, 218–237 (2010).
[CrossRef]

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

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
[CrossRef]

C. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
[CrossRef]

G. M. Dai, “Wavefront expansion basis functions and their relationships,” J. Opt. Soc. Am. A 23, 1657–1666 (2006).
[CrossRef]

G.-M. Dai, “Wavefront expansion basis functions and their relationships: errata,” J. Opt. Soc. Am. A 23, 2970–2971 (2006).
[CrossRef]

Optom. Vis. Sci.

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

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

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

W. Harris, “Wavefronts and their propagation in astigmatic systems,” Optom. Vis. Sci. 73, 606–612 (1996).
[CrossRef] [PubMed]

E. Acosta and R. Blendowske, “Paraxial propagation of astigmatic wavefronts in optical systems by an augmented stepalong method for vergences,” Optom. Vis. Sci. 82, 923–32 (2005).
[CrossRef] [PubMed]

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72 (2007).
[CrossRef]

S. Afr. Optom.

L. Thibos, “Propagation of astigmatic wavefronts using power vectors,” S. Afr. Optom. 62, 111–113 (2003).

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).
[CrossRef] [PubMed]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), pp. 109–232.

V. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics (SPIE, 1998), pp. 91–361.

H. Diepes and R. Blendowske, Optik und Technik der Brille (Optische Fachveröffentlichung GmbH, 2002), pp. 477–486.

J. Arasa and J. Alda, “Real ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120027488.
[CrossRef]

J. Alda and J. Arasa, “Paraxial ray tracing,” in Encyclopedia of Optical Engineering (Marcel Dekker, 2004), doi:101081EEOE120009643.
[CrossRef]

G. Dai, Wavefront Optics for Vision Correction (SPIE, 2008), pp. 129–255.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Propagation of a spherical wavefront w o with a vergence distance s o about the distance d to the propagated wavefront w p with a vergence distance s p .

Fig. 2
Fig. 2

Propagation of a wavefront w o about the distance d = τ n to the propagated wavefront w p .

Fig. 3
Fig. 3

Logical flow of the computation of the Zernike coefficients of a propagated wavefront for the given Zernike coefficients of the original wavefront.

Tables (6)

Tables Icon

Table 1 Zernike Coefficients of the Original and Propagated Wavefront in Example A1 a

Tables Icon

Table 2 Zernike Coefficients of the Original and Propagated Wavefront in Example A2 a

Tables Icon

Table 3 Zernike Coefficients of the Original and Propagated Wavefront in Example B1 a

Tables Icon

Table 4 Zernike Coefficients of the Original and Propagated Wavefront in Example B2 a

Tables Icon

Table 5 Local Aberrations of the Original and Propagated Wavefront (Values Based on Our Method) in Examples A1 and A2

Tables Icon

Table 6 Local Aberrations of the Original and Propagated Wavefront (Values Based on Our Method) in Examples B1 and B2

Equations (78)

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

S p = 1 1 d n S o S o ,
n ( w o ( 2 , 0 ) w o ( 1 , 1 ) w o ( 1 , 1 ) w o ( 2 , 0 ) )
S o = ( S o , x x S o , x y S o , x y S o , y y ) = n ( w o ( 2 , 0 ) w o ( 1 , 1 ) w o ( 1 , 1 ) w o ( 0 , 2 ) ) , S p = ( S p , x x S p , x y S p , x y S p , y y ) = n ( w p ( 2 , 0 ) w p ( 1 , 1 ) w p ( 1 , 1 ) w p ( 0 , 2 ) ) .
S p = 1 1 d n S o S o ,
1 = ( 1 0 0 1 ) .
w o ( y ) = ( y w o ( y ) ) ,
w o ( y ) = k = 0 a o , k k ! y k .
y n w ( y ) | y = 0 n w ( 1 ) ( 0 ) = n ( 1 ) ( 0 ) w o ( 2 ) ( 0 ) = ( 1 0 ) w o ( 2 ) ( 0 ) ,
( y o w o ( y o ) ) + τ n n w = ( y p w p ( y p ) ) .
p ( y p ) = ( y o ( y p ) w p ( y p ) )
( 0 0 ) = ( y o + τ n n w , y y p w o ( y o ) + τ n n w , z w p ( y p ) ) .
f ( p , y p ) = ( y o + τ n n w , y ( w o ( 1 ) ( y o ) ) y p w o ( y o ) + τ n n w , z ( w o ( 1 ) ( y o ) ) w p ) ,
f ( p ( y p ) , y p ) = 0 ,
j = 1 2 f i p j p j ( 1 ) ( y p ) + f i y p = 0 , i = 1 , 2 ,
A ( f 1 y o f 1 w p f 2 y o f 2 w p ) = ( 1 + τ n n w , y ( 1 ) w o ( 2 ) 0 w o ( 1 ) + τ n n w , z ( 1 ) w o ( 2 ) 1 ) .
b f y p = ( 1 0 ) .
A ( p ( y p ) ) p ( 1 ) ( y p ) = b .
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 τ n w o ( 2 ) 0 0 1 ) A ( 0 ) 1 = ( 1 1 τ n w o ( 2 ) 0 0 1 ) .
p ( 1 ) ( 0 ) = A 1 b = ( 1 1 τ n w o ( 2 ) 0 ) .
w p ( 2 ) = β w o ( 2 ) ,
β = 1 1 τ n w o ( 2 ) ,
w p ( 3 ) = β 3 w o ( 3 ) , w p ( 4 ) = β 4 ( w o ( 4 ) + 3 τ n ( β w o ( 3 ) 2 w o ( 2 ) 4 ) ) , w p ( 5 ) = β 5 ( w o ( 5 ) + 5 β τ n w o ( 3 ) ( 2 w o ( 4 ) + 3 β τ n w o ( 3 ) 2 6 w o ( 2 ) 3 ) ) , .
w p ( k ) = β k ( w o ( k ) + R k ) ,
p ( 2 ) = A 1 A ( 1 ) A 1 b , k = 2 , p ( k ) = A 1 j = 1 2 ( k 1 j 1 ) A ( k j ) p ( j ) , k 3 .
p ( 2 ) = A 1 A ( 1 ) A 1 b , k = 2 , p ( k ) = A 1 ( ( k 1 ) A ( k 2 ) A 1 A ( 1 ) A ( k 1 ) ) A 1 b , k 3.
A ( 1 ) = ( τ n 1 τ n w o ( 2 ) w o ( 3 ) 0 w o ( 2 ) 0 ) ,
p ( 3 ) = A 1 ( 2 A ( 1 ) A 1 A ( 1 ) A ( 2 ) ) A 1 b , k = 3 , p ( k ) = A 1 A ( k 1 ) A 1 b , otherwise .
A ( 2 ) = ( τ n ( τ n w o ( 3 ) + w o ( 4 ) ) 0 w o ( 3 ) 0 ) .
A ( m ) = ( τ n w o ( m + 2 ) 0 w o ( m + 1 ) 0 ) ,
w p ( k ) = w o ( k )
w o ( x , y ) = ( x y w o ( x , y ) ) ,
n ( u , v ) 1 1 + u 2 + v 2 ( u v 1 ) ,
p ( x p , y p ) = ( x o ( x p , y p ) y o ( x p , y p ) w p ( x p , y p ) ) ,
f ( p ( x p , y p ) , x p , y p ) = 0 ,
A ( p ( x p , y p ) ) p ( 1 , 0 ) ( x p , y p ) = b x , A ( p ( x p , y p ) ) p ( 0 , 1 ) ( x p , y p ) = b y ,
b x = f x p = ( 1 0 0 ) T , b y = f y p = ( 0 1 0 ) T ,
A ( p ( x p , y p ) ) = ( 1 + τ n ( n w , x ( 0 , 1 ) w o ( 1 , 1 ) + n w , x ( 1 , 0 ) w o ( 2 , 0 ) ) τ n ( n w , x ( 0 , 1 ) w o ( 0 , 2 ) + n w , x ( 1 , 0 ) w o ( 1 , 1 ) ) 0 τ n ( n w , y ( 0 , 1 ) w o ( 1 , 1 ) + n w , y ( 1 , 0 ) w o ( 2 , 0 ) ) 1 + τ n ( n w , y ( 0 , 1 ) w o ( 0 , 2 ) + n w , y ( 1 , 0 ) w o ( 1 , 1 ) ) 0 w o ( 1 , 0 ) + τ n ( n w , z ( 0 , 1 ) w o ( 1 , 1 ) + n w , z ( 1 , 0 ) w o ( 2 , 0 ) ) w o ( 0 , 1 ) + τ n ( n w , z ( 0 , 1 ) w o ( 0 , 2 ) + n w , z ( 1 , 0 ) w o ( 1 , 1 ) ) 1 ) .
A ( 0 ) = ( 1 τ n w o ( 2 , 0 ) τ n w o ( 1 , 1 ) 0 τ n w o ( 1 , 1 ) 1 τ n w o ( 0 , 2 ) 0 0 0 1 ) A ( 0 ) 1 = ( γ ( 1 τ n w o ( 0 , 2 ) τ n w o ( 1 , 1 ) τ n w o ( 1 , 1 ) 1 τ n w o ( 2 , 0 ) ) 0 0 0                                           0 1 ) ,
p ( 1 , 0 ) ( 0 , 0 ) = γ ( n ( n τ w o ( 0 , 2 ) ) n τ w o ( 1 , 1 ) 0 ) , p ( 0 , 1 ) ( 0 , 0 ) = γ ( n τ w o ( 1 , 1 ) n ( n τ w o ( 2 , 0 ) ) 0 ) .
w p ( 2 , 0 ) = γ ( τ n ( w o ( 1 , 1 ) ) 2 + ( 1 τ n w o ( 0 , 2 ) ) w o ( 2 , 0 ) ) , w p ( 1 , 1 ) = γ w o ( 1 , 1 ) , w p ( 0 , 2 ) = γ ( τ n ( w o ( 1 , 1 ) ) 2 + ( 1 τ n w o ( 2 , 0 ) ) w o ( 0 , 2 ) ) ,
w p ( 2 , 0 ) = 1 1 τ n w o ( 2 , 0 ) w o ( 2 , 0 ) , w p ( 1 , 1 ) = 0 , w p ( 0 , 2 ) = 1 1 τ n w o ( 0 , 2 ) w o ( 0 , 2 ) .
w p ( 3 , 0 ) = γ 3 ( ( 1 τ n w o ( 0 , 2 ) ) 3 w o ( 3 , 0 ) + τ n w o ( 1 , 1 ) ( 3 ( 1 τ n w o ( 0 , 2 ) ) 2 w o ( 2 , 1 ) + τ n w o ( 1 , 1 ) ( τ n w o ( 0 , 3 ) w o ( 1 , 1 ) ) + 3 ( 1 τ n w o ( 0 , 2 ) ) 2 w o ( 1 , 2 ) ) ) , w p ( 2 , 1 ) = γ 3 ( w o ( 2 , 1 ) + τ n ( w o ( 1 , 1 ) ( 2 w o ( 1 , 2 ) + w o ( 3 , 0 ) ) ( 2 w o ( 0 , 2 ) + w o ( 2 , 0 ) ) w o ( 2 , 1 ) ) + ( τ n ) 2 ( w o ( 2 , 1 ) w o ( 0 , 2 ) 2 2 ( w o ( 1 , 1 ) ( w o ( 1 , 2 ) + w o ( 3 , 0 ) ) w o ( 2 , 0 ) w o ( 2 , 1 ) ) w o ( 0 , 2 ) + w o ( 0 , 3 ) w o ( 1 , 1 ) 2 + 2 w o ( 1 , 1 ) ( w o ( 1 , 1 ) w o ( 2 , 1 ) w o ( 1 , 2 ) w o ( 2 , 0 ) ) ) + ( τ n ) 3 ( w o ( 1 , 2 ) w o ( 1 , 1 ) 3 ( w o ( 0 , 3 ) w o ( 2 , 0 ) + 2 w o ( 0 , 2 ) w o ( 2 , 1 ) ) w o ( 1 , 1 ) 2 + w o ( 0 , 2 ) ( 2 w o ( 1 , 2 ) w o ( 2 , 0 ) + w o ( 0 , 2 ) w o ( 3 , 0 ) ) w o ( 1 , 1 ) w o ( 0 , 2 ) 2 w o ( 2 , 0 ) w o ( 2 , 1 ) ) ) , w p ( 1 , 2 ) = γ 3 ( w o ( 1 , 2 ) + τ n ( w o ( 1 , 1 ) ( 2 w o ( 2 , 1 ) + w o ( 0 , 3 ) ) ( 2 w o ( 2 , 0 ) + w o ( 0 , 2 ) ) w o ( 1 , 2 ) ) + ( τ n ) 2 ( w o ( 1 , 2 ) w o ( 2 , 0 ) 2 2 ( w o ( 1 , 1 ) ( w o ( 2 , 1 ) + w o ( 0 , 3 ) ) w o ( 0 , 2 ) w o ( 1 , 2 ) ) w o ( 2 , 0 ) + w o ( 3 , 0 ) w o ( 1 , 1 ) 2 + 2 w o ( 1 , 1 ) ( w o ( 1 , 1 ) w o ( 1 , 2 ) w o ( 2 , 1 ) w o ( 0 , 2 ) ) ) + ( τ n ) 3 ( w o ( 2 , 1 ) w o ( 1 , 1 ) 3 ( w o ( 3 , 0 ) w o ( 0 , 2 ) + 2 w o ( 2 , 0 ) w o ( 1 , 2 ) ) w o ( 1 , 1 ) 2 + w o ( 2 , 0 ) ( 2 w o ( 2 , 1 ) w o ( 0 , 2 ) + w o ( 2 , 0 ) w o ( 0 , 3 ) ) w o ( 1 , 1 ) w o ( 2 , 0 ) 2 w o ( 0 , 2 ) w o ( 1 , 2 ) ) ) , w p ( 0 , 3 ) = γ 3 ( ( 1 τ n w o ( 2 , 0 ) ) 3 w o ( 0 , 3 ) + τ n w o ( 1 , 1 ) ( 3 ( 1 τ n w o ( 2 , 0 ) ) 2 w o ( 1 , 2 ) + τ n w o ( 1 , 1 ) ( τ n w o ( 3 , 0 ) w o ( 1 , 1 ) ) + 3 ( 1 τ n w o ( 2 , 0 ) ) 2 w o ( 2 , 1 ) ) ) .
p ( k x , 0 ) = A 1 A ( k x 1 , 0 ) A 1 b x , k x 3 , k y = 0 , p ( 3 , 0 ) = A 1 ( 2 A ( 1 , 0 ) A 1 A ( 1 , 0 ) A ( 2 , 0 ) ) A 1 b x , k x = 3 , k y = 0 , p ( 2 , 1 ) = A 1 ( A ( 1 , 0 ) A 1 A ( 0 , 1 ) + A ( 0 , 1 ) A 1 A ( 1 , 0 ) A ( 2 , 0 ) ) A 1 b x , k x = 2 , k y = 1 , p ( k x , k y ) = A 1 A ( k x 1 , k y ) A 1 b x , k x 0 , k y 0 , k x + k y 3 , = A 1 A ( k x , k y 1 ) A 1 b y , p ( 1 , 2 ) = A 1 ( A ( 0 , 1 ) A 1 A ( 1 , 0 ) + A ( 1 , 0 ) A 1 A ( 0 , 1 ) A ( 0 , 2 ) ) A 1 b y , k x = 1 , k y = 2 , p ( 0 , 3 ) = A 1 ( 2 A ( 0 , 1 ) A 1 A ( 0 , 1 ) A ( 0 , 2 ) ) A 1 b y , k x = 0 , k y = 3 , p ( 0 , k y ) = A 1 A ( 0 , k y 1 ) A 1 b y , k x = 0 , k y 3.
A ( m x , m y ) = { ( 0 A m x , m y 0 w o ( m x + 1 , m y + 1 ) w o ( m x , m y ) 0 ) , m x + m y = 2 , ( τ n w o ( m x + 2 , m y ) τ n w o ( m x + 1 , m y + 1 ) 0 τ n w o ( m x + 1 , m y + 1 ) τ n w o ( m x , m y + 2 ) 0 w o ( m x + 1 , m y ) w o ( m x , m y + 1 ) 0 ) , m x + m y > 2 ,
A m x , m y = τ n ( τ n v ( w o ( 3 , 0 ) w o ( 2 , 1 ) ) + w o ( m x + 2 , m y ) τ n v ( w o ( 2 , 1 ) w o ( 1 , 2 ) ) + w o ( m x + 1 , m y + 1 ) τ n v ( w o ( 2 , 1 ) w o ( 1 , 2 ) ) + w o ( m x + 1 , m y + 1 ) τ n v ( w o ( 1 , 2 ) w o ( 0 , 3 ) ) + w o ( m x , m y + 2 ) ) , v = ( w o ( m x + 1 , m y ) w o ( m x , m y + 1 ) ) ,
w p ( k x , k y ) = w o ( k x , k y )
S p = β S o , E p , 3 = β 3 E o , 3 , E p , 4 = β 4 ( E o , 4 + 3 d n ( β E o , 3 2 S o 4 n 2 ) ) , E p , 5 = β 5 ( E o , 5 + 5 β d n E o , 3 ( 2 E o , 4 + 3 β d n E o , 3 2 6 S o 3 n 2 ) ) , E p , 6 = β 6 ( E o , 6 + 5 β d n ( 3 E o , 3 E o , 5 + 21 β d n E o , 3 2 E o , 4 12 S o 3 E o , 4 n 2 + 2 E o , 4 2 9 β S 0 2 E o , 3 2 3 + 4 d n S o n 2 + 21 ( β d n ) 2 E o , 3 4 + 9 S o 6 1 + d n S o n 4 ) ) .
E p , k = β k ( E o , k + R k ) ,
E p , k = E o , k .
s p = γ ( s o + d n ( S o , x y 2 S o , x x S o , y y 0 S o , x y 2 S o , x x S o , y y ) ) ,
γ = 1 1 d n S o , x x ( d n S o , x y ) 2 d n S o , y y + ( d n ) 2 S o , x x S o , y y
e u p , 3 = ( β x x 3 3 β x x 2 β x y 3 β x x β x y 2 β x y 3 β x x 2 β x y β x x ( β x x β y y + 2 β x y 2 ) β x y ( 2 β x x β y y + β x y 2 ) β x y 2 β y y β x x β x y 2 β x y ( 2 β x x β y y + β x y 2 ) β y y ( β x x β y y + 2 β x y 2 ) β x y β y y 2 β x y 3 3 β x y 2 β y y 3 β x y β y y 2 β y y 3 ) e o , 3 ,
( β x x β x y β x y β y y ) = ( 1 τ n ( w o ( 2 , 0 ) w o ( 1 , 1 ) w o ( 1 , 1 ) w o ( 0 , 2 ) ) ) 1 = ( 1 d n ( S o , x x S o , x y S o , x y S o , y y ) ) 1 .
α = 1 2 arctan ( 2 S o , x y S o , y y S o , x x ) .
s p = ( β x x 0 0 0 0 0 0 0 β y y ) s o ,
e p , 3 = ( β x x 3 0 0 0 0 β x x 2 β y y 0 0 0 0 β x x β y y 2 0 0 0 0 β y y 3 ) e o , 3 ,
e p , 4 = ( β x x 4 0 β x x 3 β y y 1 β x x 2 β y y 2 β x x 1 β y y 3 0 β y y 4 ) , ( e o , 4 + d n ( 3 ( β x x E o , x x x 2 + β y y E o , x x y 2 S o , x x 4 n 2 ) 3 E o , x x y ( β x x E o , x x x + β y y E o , x y y ) β x x ( 2 E o , x x y 2 + E o , x x x E o , x y y ) + β y y ( 2 E o , x y y 2 + E o , x x y E o , y y y ) ( S o , x x S o , y y n ) 2 3 E o , x y y ( β x x E o , x x y + β y y E o , y y y ) 3 ( β x x E o , x y y 2 + β y y E o , y y y 2 S o , y y 4 n 2 ) ) ) .
e p , k = B k ( e o , k + r k ) ,
B k = ( β x x k 0 β x x k 1 β y y 1 β x x 1 β y y k 1 0 β y y k ) .
e p , k = e o , k ,
s p = ( β x x S o , x x 0 β y y S o , x x ) .
e p , 4 = ( β x x 4 E o , x x x x 0 β x x 2 β y y 2 E o , x x y y 0 β y y 4 E o , y y y y ) d n 3 ( 3 β x x 4 S o , x x 4 0 β x x 2 β y y 2 ( S o , x x S o , y y ) 2 0 3 β y y 4 S o , y y 4 ) ,
E p , x x x x x x = β x x 6 ( E o , x x x x x x + 5 d n 5 ( 2 β x x ( n 2 E o , x x x x 3 S o , x x 3 ) 9 S o , x x 6 ) ) , E p , x x x x y y = β x x 4 β y y 2 ( E o , x x x x y y + d n 6 ( 6 β y y n 5 E o , x x x x 2 + β x x S o , x x S o , y y 2 ( 3 S o , x x 3 ( n + d ( 3 2 β y y ) S o , x x + 2 n β y y ) 4 n 3 E o , x x x x ) + 4 n 3 β x x E o , x x y y ( n 2 E o , x x x x + 3 S o , x x 3 ( ( β y y 2 ) S o , x x β y y S o , y y ) ) ) ) , E p , x x y y y y = E p , x x x x y y ( x y ) , E p , y y y y y y = E p , x x x x x x ( x y ) .
n ( v ) 1 1 + v 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 .
p ( 1 ) ( y p ) = A ( p ( y p ) ) 1 b .
p ( 1 ) ( 0 ) = A 1 b , p ( 2 ) ( 0 ) = ( A 1 ) ( 1 ) b , , p ( k ) ( 0 ) = ( A 1 ) ( k 1 ) b ,
Ap ( 1 ) ( 0 ) = b ,
A ( 1 ) p ( 1 ) ( 0 ) + Ap ( 2 ) ( 0 ) = 0 ,
A ( 2 ) p ( 1 ) ( 0 ) + 2 A ( 1 ) p ( 2 ) ( 0 ) + Ap ( 3 ) ( 0 ) = 0 ,
j = 1 k ( k 1 j 1 ) A ( k j ) p ( j ) ( 0 ) = 0 , k 2.
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 .
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 .
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.

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