Abstract

A ray transfer matrix is used to calculate the propagation of aberrated wavefronts across a homogeneous refractive index. The wavefront is represented by local surface normals, i.e., by a ray bundle, and the propagation is accomplished by transferring those rays across the space. Wavefront shape is generated from the slopes and positions of the collection of rays. Calculation methods are developed for the paraxial case, for higher-order expansions, and for the exact tangent case. A numerical example is used to compare results between an analytical method and the methods developed here.

© 2014 Optical Society of America

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References

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    [CrossRef]
  2. W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73, 606–612 (1996).
  3. M. P. Keating, “Lens effectivity in terms of dioptric power matrices,” Am. J. Optom. Physiol. Opt. 58, 1154–1160 (1981).
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    [CrossRef]
  8. R. Blendowske and E. Acosta, “Paraxial propagation of astigmatic wavefronts through noncoaxial astigmatic optical systems,” Optom. Vis. Sci. 83, 119–122 (2006).
    [CrossRef]
  9. G. M. Dai, C. E. 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]
  10. G. Esser, W. Becken, W. Muller, P. Baumbach, J. Arasa, and D. Uttenweiler, “Derivation of the propagation equations for higher order aberrations of local wavefronts,” J. Opt. Soc. Am. A 28, 2442–2458 (2011).
    [CrossRef]
  11. A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).
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    [CrossRef]
  13. M. P. Keating, “A system matrix for astigmatic optical systems: II. Corrected systems including an astigmatic eye,” Am. J. Optom. Physiol. Opt. 58, 919–929 (1981).
  14. M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Optom. Physiol. Opt. 58, 810–819 (1981).
  15. M. P. Keating, “Advantages of a block matrix formulation for an astigmatic system,” Am. J. Optom. Physiol. Opt. 59, 851–857 (1982).
    [CrossRef]
  16. W. F. Harris, “Torsional analogue of Prentice’s equation and torsional prismatic effect in astigmatic lenses,” Ophthalmic Physiolog. Opt. 10, 203–204 (1990).
    [CrossRef]
  17. W. F. Harris, “Ray vector fields, prismatic effect, and thick astigmatic optical systems,” Optom. Vis. Sci. 73, 418–423 (1996).
    [CrossRef]
  18. W. F. Harris, “Dioptric power: its nature and its representation in three- and four-dimensional space,” Optom. Vis. Sci. 74, 349–366 (1997).
    [CrossRef]
  19. W. F. Harris, “Power vectors versus power matrices, and the mathematical nature of dioptric power,” Optom. Vis. Sci. 84, 1060–1063 (2007).
    [CrossRef]
  20. J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).
  21. E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72–E78 (2007).
    [CrossRef]
  22. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

2011

2010

2009

2007

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

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72–E78 (2007).
[CrossRef]

2006

R. Blendowske and E. Acosta, “Paraxial propagation of astigmatic wavefronts through noncoaxial astigmatic optical systems,” Optom. Vis. Sci. 83, 119–122 (2006).
[CrossRef]

2005

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

2003

L. N. Thibos, “Propagation of astigmatic wavefronts using power vectors,” South African Optometrist 62, 111–113 (2003).

2000

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

1997

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

1996

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73, 606–612 (1996).

W. F. Harris, “Ray vector fields, prismatic effect, and thick astigmatic optical systems,” Optom. Vis. Sci. 73, 418–423 (1996).
[CrossRef]

1990

W. F. Harris, “Torsional analogue of Prentice’s equation and torsional prismatic effect in astigmatic lenses,” Ophthalmic Physiolog. Opt. 10, 203–204 (1990).
[CrossRef]

1982

M. P. Keating, “Advantages of a block matrix formulation for an astigmatic system,” Am. J. Optom. Physiol. Opt. 59, 851–857 (1982).
[CrossRef]

1981

M. P. Keating, “A system matrix for astigmatic optical systems: II. Corrected systems including an astigmatic eye,” Am. J. Optom. Physiol. Opt. 58, 919–929 (1981).

M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Optom. Physiol. Opt. 58, 810–819 (1981).

M. P. Keating, “Lens effectivity in terms of dioptric power matrices,” Am. J. Optom. Physiol. Opt. 58, 1154–1160 (1981).

1976

W. F. Long, “A matrix formalism for decentration problems,” Am. J. Optom. Physiol. Opt. 53, 27–33 (1976).
[CrossRef]

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–E78 (2007).
[CrossRef]

R. Blendowske and E. Acosta, “Paraxial propagation of astigmatic wavefronts through noncoaxial astigmatic optical systems,” Optom. Vis. Sci. 83, 119–122 (2006).
[CrossRef]

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

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

Arasa, J.

Awwal, A. A. S.

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).

Baumbach, P.

Becken, W.

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–E78 (2007).
[CrossRef]

R. Blendowske and E. Acosta, “Paraxial propagation of astigmatic wavefronts through noncoaxial astigmatic optical systems,” Optom. Vis. Sci. 83, 119–122 (2006).
[CrossRef]

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

Burch, J.

A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Campbell, C. E.

Chen, L.

Chernyak, D.

Dai, G. M.

Esser, G.

Gerrard, A.

A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

Harris, W. F.

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

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

W. F. Harris, “Ray vector fields, prismatic effect, and thick astigmatic optical systems,” Optom. Vis. Sci. 73, 418–423 (1996).
[CrossRef]

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73, 606–612 (1996).

W. F. Harris, “Torsional analogue of Prentice’s equation and torsional prismatic effect in astigmatic lenses,” Ophthalmic Physiolog. Opt. 10, 203–204 (1990).
[CrossRef]

Keating, M. P.

M. P. Keating, “Advantages of a block matrix formulation for an astigmatic system,” Am. J. Optom. Physiol. Opt. 59, 851–857 (1982).
[CrossRef]

M. P. Keating, “Lens effectivity in terms of dioptric power matrices,” Am. J. Optom. Physiol. Opt. 58, 1154–1160 (1981).

M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Optom. Physiol. Opt. 58, 810–819 (1981).

M. P. Keating, “A system matrix for astigmatic optical systems: II. Corrected systems including an astigmatic eye,” Am. J. Optom. Physiol. Opt. 58, 919–929 (1981).

Lin, J.

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).

Long, W. F.

W. F. Long, “A matrix formalism for decentration problems,” Am. J. Optom. Physiol. Opt. 53, 27–33 (1976).
[CrossRef]

Muller, W.

Porter, J.

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).

Queener, H.

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

Thibos, L. N.

L. N. Thibos, “Propagation of astigmatic wavefronts using power vectors,” South African Optometrist 62, 111–113 (2003).

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

Thompson, K. P.

Thorn, K.

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).

Uttenweiler, D.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

Zhao, H.

Am. J. Optom. Physiol. Opt.

W. F. Long, “A matrix formalism for decentration problems,” Am. J. Optom. Physiol. Opt. 53, 27–33 (1976).
[CrossRef]

M. P. Keating, “A system matrix for astigmatic optical systems: II. Corrected systems including an astigmatic eye,” Am. J. Optom. Physiol. Opt. 58, 919–929 (1981).

M. P. Keating, “A system matrix for astigmatic optical systems: I. Introduction and dioptric power relations,” Am. J. Optom. Physiol. Opt. 58, 810–819 (1981).

M. P. Keating, “Advantages of a block matrix formulation for an astigmatic system,” Am. J. Optom. Physiol. Opt. 59, 851–857 (1982).
[CrossRef]

M. P. Keating, “Lens effectivity in terms of dioptric power matrices,” Am. J. Optom. Physiol. Opt. 58, 1154–1160 (1981).

Appl. Opt.

J. Opt. Soc. Am. A

J. Refract. Surg.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Report from the VSIA taskforce on standards for reporting optical aberrations of the eye,” J. Refract. Surg. 16, S654–S655 (2000).

Ophthalmic Physiolog. Opt.

W. F. Harris, “Torsional analogue of Prentice’s equation and torsional prismatic effect in astigmatic lenses,” Ophthalmic Physiolog. Opt. 10, 203–204 (1990).
[CrossRef]

Optom. Vis. Sci.

W. F. Harris, “Ray vector fields, prismatic effect, and thick astigmatic optical systems,” Optom. Vis. Sci. 73, 418–423 (1996).
[CrossRef]

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

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

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

W. F. Harris, “Wavefronts and their propagation in astigmatic optical systems,” Optom. Vis. Sci. 73, 606–612 (1996).

E. Acosta and R. Blendowske, “Paraxial optics of astigmatic systems: relations between the wavefront and the ray picture approaches,” Optom. Vis. Sci. 84, E72–E78 (2007).
[CrossRef]

R. Blendowske and E. Acosta, “Paraxial propagation of astigmatic wavefronts through noncoaxial astigmatic optical systems,” Optom. Vis. Sci. 83, 119–122 (2006).
[CrossRef]

South African Optometrist

L. N. Thibos, “Propagation of astigmatic wavefronts using power vectors,” South African Optometrist 62, 111–113 (2003).

Other

A. Gerrard and J. Burch, Introduction to Matrix Methods in Optics (Wiley, 1975).

J. Porter, H. Queener, J. Lin, K. Thorn, and A. A. S. Awwal, Adaptive Optics for Vision Science: Principles, K. Chang, ed., Wiley Series in Microwave and Optical Engineering (Wiley, 2006).

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Tables (1)

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Table 1. Propagation of Wavefront from Position 0 to Position 0.02 ma

Equations (6)

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ri=(pqxy)i,
P=(10000100d¯0100d¯01)
(10000100d¯0100d¯01)(pqxy)i=(pqd¯p+xd¯q+y)=(pqxy)j.
(10000100d¯0100d¯01)(p1p2pnq1q2qnx1x2xny1y2yn)i=(p1p2pnq1q2qnx1x2xny1y2yn)j.
(00000000d¯/30000d¯/300)((p1)3(q1)3x1y1(p2)3(q2)3x2y2(pn)3(qn)3xnyn)=(00d¯(p1)3/3d¯(q1)3/300d¯(p2)3/3d¯(q2)3/300d¯(pn)3/3d¯(qn)3/3).
(10000100d¯0100d¯01)(tan(p1)tan(q1)x1y1tan(pn)tan(qn)xnyn)i=(tan(p1)tan(q1)d¯tan(p1)+x1d¯tan(q1)+y1tan(pn)tan(qn)d¯tan(pn)+xnd¯tan(qn)+yn)j.

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