Abstract

Local wavefront curvature transformations at an arbitrarily shaped optical surface are commonly determined by generalized Coddington equations that are developed here via a local thin optical element approximation. Eikonal distributions of the incident and refracted beams are calculated and related by an eikonal transfer function of a local thin optical element located in close proximity to a given point at a tangent plane of an optical surface. Main coefficients and terms involved in the generalized Coddington equations are derived and explained as a local nonparaxial generalization for the customary paraxial wavefront transformations.

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References

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  1. R. Kingslake, “Who discovered Coddington's equations?” Opt. Photonics News, May 1994, pp. 20-23.
  2. W. J. Smith, Modern Optical Engineering (McGraw-Hill Professional Publishing, 2000),Section 10.6.
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    [CrossRef]
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  17. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), Sec. 5.1.
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2008 (1)

2007 (1)

2006 (1)

2000 (2)

M. Testorf, “On the zero-thickness model of diffractive optical elements,” J. Opt. Soc. Am. A 17, 1132-1133 (2000).
[CrossRef]

S. A. Comastri and J. M. Simon, “Wavefront aberration function: its first and second field derivatives,” Optik 111, 249-260 (2000).

1999 (1)

1996 (1)

1993 (1)

1992 (1)

1986 (1)

1981 (1)

1976 (1)

1964 (1)

1957 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999), Sec. 4.6.
[PubMed]

Burkhard, D. G.

Campbell, C. E.

Comastri, S. A.

S. A. Comastri and J. M. Simon, “Wavefront aberration function: its first and second field derivatives,” Optik 111, 249-260 (2000).

DeJager, D.

Giusfredi, G.

Golub, M. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), Sec. 5.1.

Greco, V.

Kingslake, R.

R. Kingslake, “Who discovered Coddington's equations?” Opt. Photonics News, May 1994, pp. 20-23.

Kneisly II, J. A.

Landau, L. D.

L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields, Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 2003), Vol. 2, p. 159.

Landgrave, J. E. A.

Lifschitz, E. M.

L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields, Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 2003), Vol. 2, p. 159.

Lindlein, N.

Moya-Cessa, J. R.

Murray, A. E.

Noethen, M.

Rolland, J. P.

Schwider, J.

Shealy, D. L.

Simon, J. M.

S. A. Comastri and J. M. Simon, “Wavefront aberration function: its first and second field derivatives,” Optik 111, 249-260 (2000).

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill Professional Publishing, 2000),Section 10.6.

Stavroudis, O. N.

O. N. Stavroudis, “Simpler derivation of the formulas for generalized ray tracing,” J. Opt. Soc. Am. 66, 1330-1333 (1976).
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-179.

Testorf, M.

Thompson, K. P.

Turunen, J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999), Sec. 4.6.
[PubMed]

Zou, W.

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Optik (1)

S. A. Comastri and J. M. Simon, “Wavefront aberration function: its first and second field derivatives,” Optik 111, 249-260 (2000).

Other (6)

L. D. Landau and E. M. Lifschitz, The Classical Theory of Fields, Course of Theoretical Physics, 4th ed. (Butterworth-Heinemann, 2003), Vol. 2, p. 159.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005), Sec. 5.1.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999), Sec. 4.6.
[PubMed]

R. Kingslake, “Who discovered Coddington's equations?” Opt. Photonics News, May 1994, pp. 20-23.

W. J. Smith, Modern Optical Engineering (McGraw-Hill Professional Publishing, 2000),Section 10.6.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), pp. 136-179.

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Figures (5)

Fig. 1
Fig. 1

Local coordinate system at an optical surface; ζ ¯ follows normal N ¯ . Axes u ¯ , v ¯ , are associated with the plane of incidence, ξ ¯ , η ¯ are principal axes, O ¯ ξ , O ¯ η are principal centers of curvature.

Fig. 2
Fig. 2

Local thin optical element model of an optical surface, where the incident beam is virtually transformed at the tangent plane of the optical surface instead of being transformed within an entire layer thickness of the optical element.

Fig. 3
Fig. 3

Coordinates and vectors at the plane of incidence. N , ζ show direction of the incident beam, N , ζ direction of the output beam, N ¯ , ζ ¯ normal to the optical surface. Axes u, u , u ¯ are perpendicular to the plane of incidence; axes v, v , v ¯ are at the plane of incidence perpendicular to N, N , N ¯ , respectively.

Fig. 4
Fig. 4

Geometry of the incident wavefront and tangent planes.

Fig. 5
Fig. 5

Local wavefront transformation at a point r of the refractive optical surface with principal directions ξ ¯ , η ¯ and axes u ¯ , v ¯ associated with the plane of incidence. Incident and output beams have principal directions ξ , η and ξ , η , distances s ξ , s η and s ξ , s η from a point r to the local principal centers of curvature. Wavefronts of the incident and output beams have a common point r with the optical surface but are shown separated for clarity.

Equations (34)

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n N = n N + μ N ¯ ,
μ = n N N ¯ n N N ¯ = n cos θ n cos θ ,
cos θ = N N ¯ , cos θ = N N . ¯
μ = { n 2 n 2 [ 1 ( N N ¯ ) 2 ] } 1 2 n N N ¯ = { n 2 n 2 sin 2 θ } 1 2 n cos θ
μ = 2 n N N ¯ = 2 n cos θ ,
ζ ¯ = c ¯ ξ ξ ¯ 2 2 + c ¯ η η ¯ 2 2 ,
ξ ¯ = u ¯ cos α ¯ v ¯ sin α ¯ ,
η ¯ = u ¯ sin α ¯ + v ¯ cos α ¯
ζ ¯ = c ¯ u u ¯ 2 2 + c ¯ v v ¯ 2 2 + τ ¯ u ¯ v ¯ ,
c ¯ u = c ¯ ξ cos 2 α ¯ + c ¯ η sin 2 α ¯ ,
c ¯ v = c ¯ ξ sin 2 α ¯ + c ¯ η cos 2 α ¯ ,
τ ¯ = 1 2 ( c ¯ ξ + c ¯ η ) sin 2 α ¯
Δ S = ( n N n N ) ( ζ ¯ N ¯ ) = μ ζ ¯ ,
S = S + Δ S = S μ ζ ¯ .
ζ lin = v ¯ sin θ ,
u = u ¯ , v = v ¯ cos θ .
S = n ( ζ N ) N = n ζ = n ( ζ lin ζ W ) = n ( v ¯ sin θ ζ W ) .
ζ W = c ξ ξ 2 2 + c η η 2 2 ,
ζ W = c u u 2 2 + c v v 2 2 + τ u v .
c u = c ξ cos 2 α + c η sin 2 α ,
c v = c ξ sin 2 α + c η cos 2 α ,
τ = 1 2 ( c ξ + c η ) sin 2 α
c ξ = c u cos 2 α + c v sin 2 α τ sin 2 α ,
c μ = c u sin 2 α + c v cos 2 α + τ sin 2 α ,
tan 2 α = 2 τ c u + c v ,
c ξ = c u cos 2 α + c v sin 2 α τ sin 2 α ,
c μ = c u sin 2 α + c v cos 2 α + τ sin 2 α ,
tan 2 α = 2 τ c u + c v .
S = n ( v ¯ sin θ c u u ¯ 2 2 c v cos 2 θ v ¯ 2 2 τ cos θ u ¯ v ¯ ) .
S = n ( v ¯ sin θ c u u ¯ 2 2 c v cos 2 θ v ¯ 2 2 τ cos θ u ¯ v ¯ ) .
n sin θ = n sin θ ,
n c u = n c u + μ c ¯ u ,
n c v cos 2 θ = n c v cos 2 θ + μ c ¯ v ,
n τ cos θ = n τ cos θ + μ τ ¯ .

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