Abstract

Using the condition of constant optical path length for rays passing through an optical system, an equation for the wave front is presented in a simplified form. The wave front equation has been explicitly evaluated for a plane wave incident on a spherical reflector or a plano-convex lens. Then, the principal radii of curvature of the reflected or refracted wave front, evaluated directly from the wave front equation, are shown to locate the caustic surfaces of the optical system. From the wave front equation, a closed form expression for the wave aberration function for a plane wave reflected by a spherical mirror or a plano-corvex lens has been evaluated and compared to the results obtained from third-order aberration theory.

© 1988 Optical Society of America

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References

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  1. M. Born, E. WolfPrinciples of Optics (Pergamon, New York, 1959), pp. 108–29.
  2. J. L. Synge, W. Corway, The Mathematical Papers of Hamilton (Cambridge U. P., London, 1931).
  3. V. A. Fock, Electromagnetic Diffraction and Propagation Problem (Pergamon, New York, 1960), p. 160.
  4. O. N. Stavroudis, R. C. Fronczek, “Caustic Surfaces and the Structure of the Geometrical Optics,” J. Opt. Soc. Am. 66, 795 (1976).
    [CrossRef]
  5. O. N. Stavroudis, “Tracing Wavefronts: Can it be Done?,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 18 (1987).
  6. J. A. Kneisly, “Local Curvature of Wave Fronts in an Optical System,” J. Opt. Soc. Am. 54, 229 (1964).
    [CrossRef]
  7. J. M. Rebordao, M. Grosmann, “Refraction on Spherical Surfaces. I: an Exact Algebraic Approach,” J. Opt. Soc. Am. A 1, 51 (1984).
    [CrossRef]
  8. L. Seidel, “On Dioptics on the Development of Third Order Coefficients,” Astron. Nachr. 43, 289 (1956).
    [CrossRef]
  9. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), p. 35.
  10. P. J. Sands, “Off-Axis Aberration Coefficients,” Thesis, Australian National University, Australia (1967).
  11. W. T. Welford, “A New Total Aberration Formula,” Opt. Acta 19, No. 9, 719 (1972).
    [CrossRef]
  12. H. H. Hopkins, Wave Theory of Aberrations (Oxford U.P., London, 1950), Chap. 1.
  13. W. T. Welford, Aberration of the Symmetrical Optical System (Academic, New York, 1974), pp. 4–45.
  14. E. Kreyszig, Introduction to Differential Geometry and Riemannian (U. Toronto Press, Toronto, Ont., 1968), pp. 57–99.
  15. D. G. Burkhard, D. L. Shealy, “Formula for the Density of Tangent Rays over a Caustic Surface,” Appl. Opt. 21, 3299 (1982).
    [CrossRef] [PubMed]
  16. O. N. Stavroudis, The Optics of Rays, Wave Front and Caustics (Academic, New York, 1972), p. 157.
  17. B. R. Nijboer, “Title,” Physica 10, 679 (1943).
    [CrossRef]
  18. E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, MA, 1963), Chap. 4.
  19. D. L. Shealy, D. G. Burkhard, “Analytical Illuminance Calculation in a Multi-Surface Optical System,” Opt. Acta 22, 485 (1975).
    [CrossRef]
  20. W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).
  21. M. Schwartz, S. Green, W. A. Rutledge, Vector Analysis (Harper & Row, New York, 1969).
  22. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 173.
  23. M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958), pp. 299–379.
  24. SYMBOLIC, Inc., Eleven Cambridge Center, Cambridge, MA 02142.

1987 (1)

O. N. Stavroudis, “Tracing Wavefronts: Can it be Done?,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 18 (1987).

1984 (1)

1982 (1)

1976 (1)

1975 (1)

D. L. Shealy, D. G. Burkhard, “Analytical Illuminance Calculation in a Multi-Surface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

1972 (1)

W. T. Welford, “A New Total Aberration Formula,” Opt. Acta 19, No. 9, 719 (1972).
[CrossRef]

1964 (1)

1956 (1)

L. Seidel, “On Dioptics on the Development of Third Order Coefficients,” Astron. Nachr. 43, 289 (1956).
[CrossRef]

1943 (1)

B. R. Nijboer, “Title,” Physica 10, 679 (1943).
[CrossRef]

Born, M.

M. Born, E. WolfPrinciples of Optics (Pergamon, New York, 1959), pp. 108–29.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), p. 35.

Burkhard, D. G.

D. G. Burkhard, D. L. Shealy, “Formula for the Density of Tangent Rays over a Caustic Surface,” Appl. Opt. 21, 3299 (1982).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Analytical Illuminance Calculation in a Multi-Surface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

Corway, W.

J. L. Synge, W. Corway, The Mathematical Papers of Hamilton (Cambridge U. P., London, 1931).

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problem (Pergamon, New York, 1960), p. 160.

Fronczek, R. C.

Green, S.

M. Schwartz, S. Green, W. A. Rutledge, Vector Analysis (Harper & Row, New York, 1969).

Grosmann, M.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958), pp. 299–379.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U.P., London, 1950), Chap. 1.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 173.

Kneisly, J. A.

Kreyszig, E.

E. Kreyszig, Introduction to Differential Geometry and Riemannian (U. Toronto Press, Toronto, Ont., 1968), pp. 57–99.

Nijboer, B. R.

B. R. Nijboer, “Title,” Physica 10, 679 (1943).
[CrossRef]

O’Neil, E. L.

E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, MA, 1963), Chap. 4.

Rebordao, J. M.

Rutledge, W. A.

M. Schwartz, S. Green, W. A. Rutledge, Vector Analysis (Harper & Row, New York, 1969).

Sands, P. J.

P. J. Sands, “Off-Axis Aberration Coefficients,” Thesis, Australian National University, Australia (1967).

Schwartz, M.

M. Schwartz, S. Green, W. A. Rutledge, Vector Analysis (Harper & Row, New York, 1969).

Seidel, L.

L. Seidel, “On Dioptics on the Development of Third Order Coefficients,” Astron. Nachr. 43, 289 (1956).
[CrossRef]

Shealy, D. L.

D. G. Burkhard, D. L. Shealy, “Formula for the Density of Tangent Rays over a Caustic Surface,” Appl. Opt. 21, 3299 (1982).
[CrossRef] [PubMed]

D. L. Shealy, D. G. Burkhard, “Analytical Illuminance Calculation in a Multi-Surface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

Stavroudis, O. N.

O. N. Stavroudis, “Tracing Wavefronts: Can it be Done?,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 18 (1987).

O. N. Stavroudis, R. C. Fronczek, “Caustic Surfaces and the Structure of the Geometrical Optics,” J. Opt. Soc. Am. 66, 795 (1976).
[CrossRef]

O. N. Stavroudis, The Optics of Rays, Wave Front and Caustics (Academic, New York, 1972), p. 157.

Synge, J. L.

J. L. Synge, W. Corway, The Mathematical Papers of Hamilton (Cambridge U. P., London, 1931).

Welford, W. T.

W. T. Welford, “A New Total Aberration Formula,” Opt. Acta 19, No. 9, 719 (1972).
[CrossRef]

W. T. Welford, Aberration of the Symmetrical Optical System (Academic, New York, 1974), pp. 4–45.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 173.

Wolf, E.

M. Born, E. WolfPrinciples of Optics (Pergamon, New York, 1959), pp. 108–29.

Appl. Opt. (1)

Astron. Nachr. (1)

L. Seidel, “On Dioptics on the Development of Third Order Coefficients,” Astron. Nachr. 43, 289 (1956).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (2)

W. T. Welford, “A New Total Aberration Formula,” Opt. Acta 19, No. 9, 719 (1972).
[CrossRef]

D. L. Shealy, D. G. Burkhard, “Analytical Illuminance Calculation in a Multi-Surface Optical System,” Opt. Acta 22, 485 (1975).
[CrossRef]

Physica (1)

B. R. Nijboer, “Title,” Physica 10, 679 (1943).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

O. N. Stavroudis, “Tracing Wavefronts: Can it be Done?,” Proc. Soc. Photo-Opt. Instrum. Eng. 766, 18 (1987).

Other (15)

M. Born, E. WolfPrinciples of Optics (Pergamon, New York, 1959), pp. 108–29.

J. L. Synge, W. Corway, The Mathematical Papers of Hamilton (Cambridge U. P., London, 1931).

V. A. Fock, Electromagnetic Diffraction and Propagation Problem (Pergamon, New York, 1960), p. 160.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), p. 35.

P. J. Sands, “Off-Axis Aberration Coefficients,” Thesis, Australian National University, Australia (1967).

E. L. O’Neil, Introduction to Statistical Optics (Addison-Wesley, Reading, MA, 1963), Chap. 4.

O. N. Stavroudis, The Optics of Rays, Wave Front and Caustics (Academic, New York, 1972), p. 157.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U.P., London, 1950), Chap. 1.

W. T. Welford, Aberration of the Symmetrical Optical System (Academic, New York, 1974), pp. 4–45.

E. Kreyszig, Introduction to Differential Geometry and Riemannian (U. Toronto Press, Toronto, Ont., 1968), pp. 57–99.

W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966).

M. Schwartz, S. Green, W. A. Rutledge, Vector Analysis (Harper & Row, New York, 1969).

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1957), p. 173.

M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958), pp. 299–379.

SYMBOLIC, Inc., Eleven Cambridge Center, Cambridge, MA 02142.

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Figures (8)

Fig. 1
Fig. 1

Refracted wave front configuration for a general optical system.

Fig. 2
Fig. 2

On-axis reflection of plane wave by a spherical reflector.

Fig. 3
Fig. 3

Principal curvatures of the reflected wave front as a function of the entrance pupil radius for on-axis incidence on a spherical reflector.

Fig. 4
Fig. 4

On-axis refraction of plane wave by a plano-convex lens.

Fig. 5
Fig. 5

Principal curvatures of the refracted wave front as a function of the normalized pupil radius for on-axis incidence on a plano-convex lens.

Fig. 6
Fig. 6

Wave aberration function for off-axis incidence on a spherical reflector.

Fig. 7
Fig. 7

Wave aberration function for on-axis incidence on a spherical reflector as a function of entrance pupil radius.

Fig. 8
Fig. 8

Comparison of the normalized value of the wave aberration function, computed by the conventional definition (a) and by the results of third-order aberration theory (b) as a function of the normalized entrance pupil radius for off-axis incidence on a spherical reflector.

Equations (55)

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X W = X N + δ ( x p , y p , α ) A ˆ N ,
A N = γ N A ˆ N 1 + ( γ N cos θ N + cos θ N ) N ˆ N ,
n N δ ( x p , y p α ) + ψ ( x p , y p , α ) = ψ 0 ,
ψ ( x p , y p ; α ) = N 1 k = 0 n k r k ( x p , y p ; α ) ,
X w = X N + [ ψ 0 k = 0 N 1 n k r k ( x p , y p ; α ) ] A ˆ N / n N .
A ˆ ( θ , ϕ ) = cos ϕ sin 2 θ i ˆ sin ϕ sin 2 θ j ˆ cos 2 θ k ˆ ,
X m ( θ , ϕ ) = R sin θ cos θ i ˆ + R sin θ sin ϕ j + R cos ϕ k ˆ .
X w ( θ , ϕ ) = R [ sin θ ( 1 cos θ ) sin 2 θ ] ( cos θ i ˆ + sin ϕ j ˆ ) + R [ cos θ ( 1 cos θ ) cos 2 θ ] k ˆ .
g θ θ = X w θ · X w θ = R 2 ( 3 cos θ 2 ) 2 ;
g ϕ ϕ = X w θ · X w θ = R 2 [ sin θ sin 2 θ ( 1 cos θ ) ] ;
g θ ϕ = X w θ · X w θ = 0 .
b θ θ = A ˆ · 2 X w θ 2 = 2 R ( 3 cos θ 2 ) ,
b ϕ ϕ = A ˆ · 2 X w θ 2 = R sin θ sin 2 θ ( 1 2 cos θ + 2 cos 2 θ ) ,
b θ ϕ = A ˆ · 2 X w θ ϕ = 0 .
K θ = b θ θ g θ θ = 2 R ( 3 cos θ 2 ) ,
K ϕ = b ϕ ϕ g ϕ ϕ = 2 c o s θ / [ R ( 1 2 cos θ + 2 cos 2 θ ) ] .
K θ = K ϕ = 2 / R .
A ˆ ( θ , ϕ ) = cos θ sin ( ϕ θ ) i ˆ sin ϕ sin ( ϕ θ ) j + cos ( ϕ θ ) k ˆ .
ψ 0 = R + ( n 1 ) d ,
ψ = R + ( n 1 ) d n R ( 1 cos θ ) ,
X w ( θ , ϕ ) = R [ sin θ n ( 1 cos θ ) sin ( ϕ θ ) ] [ cos θ i ˆ + sin ϕ j ˆ ] + R [ cos θ + n ( 1 cos θ ) cos ( θ θ ) ] k ˆ .
g θ θ = R 2 [ cos θ + n ( cos θ 1 ) / cos θ ] ,
g ϕ ϕ = R 2 [ sin θ n ( 1 cos θ ) sin ( ϕ θ ) ] 2 ,
g θ ϕ = 0 .
b θ θ = sec θ R [ n sec 2 θ cos ( θ θ ) × sec 2 θ ( n cos θ cos θ 1 ) ( 1 sec θ ) ] ,
b ϕ ϕ = r sin ( θ θ ) [ 1 + n ( 1 cos θ ) ( cos θ n cos θ ) ] ,
b θ ϕ = 0 .
K θ = b θ θ g θ θ = n cos θ cos θ R [ cos 2 θ + n ( cos θ 1 ) ( n cos θ cos θ ) ] ,
K ϕ = b ϕ ϕ g ϕ ϕ = n cos θ cos θ R [ 1 + n ( 1 cos θ ) ( n cos θ cos θ ) ] .
K θ = K ϕ = n 1 R .
X c = X m + r c A ˆ ,
X c = X m + r c A ˆ ,
r c = | X m X c | = δ + 1 / K θ ,
r c = | X m X c | = δ + 1 / K ϕ ,
δ = | X m X w | = R ( 1 cos θ ) .
r c = ( R / 2 ) cos θ ,
r c = R / ( 2 cos θ ) .
δ = n R ( 1 cos θ ) .
r c = R cos 2 θ / ( n cos θ cos θ ) ,
r c = R / ( n cos θ cos θ ) .
A ˆ 0 = sin α i ˆ + cos α k ˆ .
A ˆ ( θ , ϕ ) = sin ( 2 θ + α ) ( cos ϕ i ˆ + sin ϕ j ˆ ) cos ( 2 θ + α ) k ˆ .
X w ( θ , ϕ ) = X m ( θ , ϕ ) + ( ψ 0 r ) A ˆ ( θ , ϕ ) ,
X m ( θ , ϕ ) = R ( sin θ cos θ i ˆ + sin θ sin θ j ˆ + cos θ k ˆ ) .
ψ 0 r = R [ cos α cos ( α + θ ) ] .
X w ( θ , ϕ ) = R { sin θ [ cos α cos ( α + θ ) ] sin ( 2 θ + α ) } * ( cos θ i ˆ + sin ϕ j ˆ ) + R { cos θ [ cos α cos ( α + θ ) ] cos ( 2 θ + α ) } k ˆ .
Δ ( x p , y p ; α ) = | X w ( x p , y p ; α ) X R ( x p , y p ; α ) | ,
Δ ( x p , y p ; α ) = ( Z w Z R ) / A z ,
X R R sin θ cos ϕ Z R R cos θ = A X A Z = tan ( 2 θ + α ) cos ϕ ;
Y R R sin θ sin ϕ Z R R cos θ = A Y A Z = tan ( 2 θ + α ) sin ϕ ,
( X R + R 2 tan α ) 2 + Y R 2 + ( Z R R 2 ) 2 = ( R 2 sec α ) 2 .
Z R 2 + R [ sin θ sin ( 4 θ + 2 α ) 2 cos θ sin 2 ( 2 θ + α ) + tan α 2 sin ( 4 θ + 2 α ) cos 2 ( 2 θ + α ) ] Z R + R 2 [ cos ( 4 θ + 2 α ) × ( sin θ cos ϕ tan α + sin 2 θ ) + ( 1 sin 2 θ ) sin 2 ( 2 θ + α ) ] .
Z R = R ( B + B 2 L 2 ) / 2 ,
B = sin θ sin ( 4 θ + 2 α ) 2 cos θ sin 2 ( 2 θ + a ) + tan α 2 sin ( 4 θ + 2 α ) cos ϕ cos 2 ( 2 θ + α ) , L = 4 sin ( θ + α ) [ sin ( θ + α ) tan α cos 2 ( 2 θ + a ) cos ϕ ] .
Δ = R cos ( 2 θ + α ) { cos θ [ cos α cos ( α + θ ) ] cos ( 2 θ + α ) + ( B B 2 L 2 ) / 2 } .

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