Abstract

The equations for the refraction operation in generalized ray tracing are rederived using the Frenet equations from metric differential geometry and the idea of the directional derivative. Expressions are obtained for the principal directions and curvatures of a refracted wave front in terms of those quantities on the incident wave front and on the refracting surface. This provides the geometrical properties of the refracted wave front in the neighborhood of the traced ray.

© 1976 Optical Society of America

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References

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  1. S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” MS thesis (University of Arizona, Tucson, 1971); (1971).
  2. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chaps. IX and X.
  3. Allvar Gullstrand, “Die reelle optische Abbildung,” Sv. Vetensk. Handl. 41, 1–119 (1906).
  4. O. Altrichter and G. Schäfer, “Herleitung der Gullstrandschen Grundgleichungen für schiefe Strahlenbüschel aus den Hauptkrümmungen der Wellenfläsche,” Optik (Stuttg.) 13, 241–253 (1956).
  5. J. A. Kneisly, “Local curvature of wave fronts in an optical system,” J. Opt. Soc. Am. 54, 229–235 (1964).
    [Crossref]
  6. O. N. Stavroudis, “Ray tracing formulas for uniaxial crystals,” J. Opt. Soc. Am. 52, 187–191 (1962).
    [Crossref]
  7. Allvar Gullstrand, “Optical Imagery,” Appendix I, in Helmholtz’s Treatise on Physiological Optics, edited by J. P. C. Southall (Dover, New York, 1962), Vol. I, Part I, transl. from the 3rd German edition.
  8. D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic paraboloid, and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
    [Crossref] [PubMed]
  9. D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
    [Crossref]
  10. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometric image,” J. Opt. Soc. Am. 66, 795–800 (1976).
    [Crossref]
  11. H. A. Buchdahl, “Systems without: symmetries: Foundations of a theory of Lagrangian aberration coefficients,” J. Opt. Soc. Am. 62, 1314–1324 (1972).
    [Crossref]

1976 (1)

1975 (1)

D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[Crossref]

1973 (1)

1972 (1)

1964 (1)

1962 (1)

1956 (1)

O. Altrichter and G. Schäfer, “Herleitung der Gullstrandschen Grundgleichungen für schiefe Strahlenbüschel aus den Hauptkrümmungen der Wellenfläsche,” Optik (Stuttg.) 13, 241–253 (1956).

1906 (1)

Allvar Gullstrand, “Die reelle optische Abbildung,” Sv. Vetensk. Handl. 41, 1–119 (1906).

Altrichter, O.

O. Altrichter and G. Schäfer, “Herleitung der Gullstrandschen Grundgleichungen für schiefe Strahlenbüschel aus den Hauptkrümmungen der Wellenfläsche,” Optik (Stuttg.) 13, 241–253 (1956).

Buchdahl, H. A.

Burkhard, D. G.

Fronczek, R. C.

Gullstrand, Allvar

Allvar Gullstrand, “Die reelle optische Abbildung,” Sv. Vetensk. Handl. 41, 1–119 (1906).

Allvar Gullstrand, “Optical Imagery,” Appendix I, in Helmholtz’s Treatise on Physiological Optics, edited by J. P. C. Southall (Dover, New York, 1962), Vol. I, Part I, transl. from the 3rd German edition.

Kneisly, J. A.

Parker, S. C.

S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” MS thesis (University of Arizona, Tucson, 1971); (1971).

Schäfer, G.

O. Altrichter and G. Schäfer, “Herleitung der Gullstrandschen Grundgleichungen für schiefe Strahlenbüschel aus den Hauptkrümmungen der Wellenfläsche,” Optik (Stuttg.) 13, 241–253 (1956).

Shealy, D. L.

Stavroudis, O. N.

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

Opt. Acta (1)

D. L. Shealy and D. G. Burkhard, “Analytical illuminance calculation in a multi-interface optical system,” Opt. Acta 22, 485–501 (1975).
[Crossref]

Optik (Stuttg.) (1)

O. Altrichter and G. Schäfer, “Herleitung der Gullstrandschen Grundgleichungen für schiefe Strahlenbüschel aus den Hauptkrümmungen der Wellenfläsche,” Optik (Stuttg.) 13, 241–253 (1956).

Sv. Vetensk. Handl. (1)

Allvar Gullstrand, “Die reelle optische Abbildung,” Sv. Vetensk. Handl. 41, 1–119 (1906).

Other (3)

S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” MS thesis (University of Arizona, Tucson, 1971); (1971).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), Chaps. IX and X.

Allvar Gullstrand, “Optical Imagery,” Appendix I, in Helmholtz’s Treatise on Physiological Optics, edited by J. P. C. Southall (Dover, New York, 1962), Vol. I, Part I, transl. from the 3rd German edition.

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Equations (37)

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N × N ¯ = μ ( N × N ¯ )
sin r = μ sin i .
N = μ N ¯ + γ N ¯ ,
γ = cos r - μ cos i .
P = ( N × N ¯ ) / sin i = ( N × N ¯ ) / sin r .
Q = N × P ,             Q ¯ = N ¯ × P ,             Q = N × P .
P = Q × N ,             N = P × Q ,             Q = N × P .
cos θ = P · T .
1 / ρ p = ( cos 2 θ ) / ρ ξ + ( sin 2 θ ) / ρ η , 1 / ρ q = ( sin 2 θ ) / ρ ξ + ( cos 2 θ ) / ρ η , 1 / σ = 1 2 ( 1 / ρ ξ - 1 / ρ η ) sin 2 θ .
t = n × b ,             n = b × t ,             b = t × n .
( t · ) t = n / ρ , ( t · ) n = - t / ρ + b / τ , ( t · ) b = - n / τ ,
( t p · ) t p = ( P · ) P = n p / ρ p = N / ρ p , ( t p · ) n p = ( P · ) N = - t p / ρ p + b p / τ p = - P / ρ p + Q / σ , ( t p · ) b p = - ( P · ) Q = - n p / τ p = N / σ .
( P · ) P = N / ρ p , ( P · ) N = - P / ρ p + Q / σ , ( P · ) Q = - N / σ .
( Q · ) Q = N / ρ q , ( Q · ) N = - Q / ρ q + P / σ , ( Q · ) P = - N / σ .
N = N ¯ cos i + Q ¯ sin i , Q = - N ¯ sin i + Q ¯ cos i .
N = N ¯ cos r + Q ¯ sin r , Q = - N ¯ sin r + Q ¯ cos r .
- N × Q ¯ = Q × N ¯ = P cos i , - N × Q ¯ = Q × N ¯ = P cos r
Q ¯ = N sin i + Q cos i = N sin r + Q cos r .
( N · ) N = 0 ,             ( N · ) N = 0.
[ ( P · ) N ] × N ¯ + N × [ ( P · ) N ¯ ] = μ [ ( P · ) N ] × N ¯ + μ N × [ ( P · ) N ¯ ] .
( - P / ρ p + Q / σ ) × N ¯ + N × ( - P / ρ ¯ p + Q ¯ / σ ¯ ) = μ ( - P / ρ p + Q / σ ) × N ¯ + μ N × ( - P / ρ ¯ p + Q ¯ / σ ¯ ) .
Q ¯ / ρ p + P ( cos r ) / σ - Q / ρ ¯ p - P ( cos r ) / σ ¯ = μ Q ¯ / ρ p + μ P ( cos i ) / σ - μ Q / ρ ¯ p - μ P ( cos i ) / σ ¯ .
( cos r ) / σ - ( cos r ) / σ ¯ = μ ( cos i ) / σ - μ ( cos i ) / σ ¯
1 / ρ p - ( cos r ) / ρ ¯ p = μ / ρ p - μ ( cos i ) / ρ ¯ p .
( cos r ) / σ = μ ( cos i ) / σ + γ / σ ¯
1 / ρ p = μ / ρ p + γ / ρ ¯ p .
[ ( Q ¯ · ) N ] × N ¯ + N × [ ( Q ¯ · ) N ¯ ] = μ [ ( Q ¯ · ) N ] × N ¯ + μ N × [ ( Q ¯ · ) N ¯ ] .
( Q ¯ · ) N = sin r ( N · ) N + cos r ( Q · ) N = cos r ( - Q / ρ q + P / σ ) .
[ ( Q ¯ · ) N ] × N ¯ = cos ( - P cos / ρ q - Q ¯ / σ ) .
[ ( Q ¯ · ) N ] × N ¯ = cos i [ - P ( cos i ) / ρ q - Q ¯ / σ ] .
N × [ ( Q ¯ · ) N ¯ ] = P ( cos r ) / ρ ¯ q + Q / σ ¯ , N × [ ( Q ¯ · ) N ¯ ] = P ( cos i ) / ρ ¯ q + Q / σ ¯ .
cos r [ - P ( cos r ) / ρ q - Q ¯ / σ ] + [ P ( cos r ) / ρ ¯ q + Q / σ ¯ ] = μ cos i [ - P ( cos i ) / ρ q - Q / σ ] + [ P ( cos i ) / ρ ¯ q + Q / σ ¯ ] .
( cos 2 r ) / ρ q = μ ( cos 2 i ) / ρ q + γ / ρ ¯ q .
1 / ρ p = μ / ρ p + γ / ρ ¯ p , ( cos r ) / σ = μ ( cos i ) / σ + γ / σ ¯ , ( cos 2 r ) / ρ q = μ ( cos 2 i ) / ρ q + γ / ρ ¯ q ,
tan 2 θ = ( 2 / σ ) ( 1 / ρ q - 1 / ρ p ) .
T ξ = P cos θ + Q sin θ ,
1 / ρ ξ = ( cos 2 ξ ) / ρ p + ( sin 2 θ ) / ρ q + ( sin 2 ξ ) / σ , 1 / ρ η = ( sin 2 θ ) / ρ p + ( cos 2 θ ) / ρ q - ( sin 2 θ ) / σ .