Abstract

The aim of this paper is to obtain expressions for the k-function, the wavefront train, and the caustic associated with the light rays refracted by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in a three-dimensional homogeneous optical medium. The general results are applied to a parabolic refracting surface. For this case, we find that when the point light source is off the optical axis, the caustic locally has singularities of the hyperbolic umbilic type, while the refracted wavefront, at the caustic region, locally has singularities of the cusp ridge and swallowtail types.

© 2012 Optical Society of America

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  8. E. Romań-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustic of a spherical wave reflected by an arbitrary smooth surface,” J. Opt. Soc. Am. A 26, 2295–2305 (2009).
    [CrossRef]
  9. E. Romań-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 12, 055705 (2011).
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    [CrossRef]
  11. Stalzer and J. Henry, “Comment on the caustic curve of a parabola,” Appl. Opt. 4, 1205–1206 (1965).
    [CrossRef]
  12. I. Lazar and L. A. DeAcetis, “Rays from a parabolic reflector for an off focal-points source,” Am. J. Phys. 36, 139–141 (1968).
    [CrossRef]
  13. R. B. Harris, “Rays from a parabolic reflector for an off focal-point source,” Am. J. Phys. 36, 1022–1023 (1968).
    [CrossRef]
  14. L. A. DeAcetis, “Wavefronts from a parabolic reflector for an off-focal source and comment on the off-axis caustic,” Am. J. Phys. 36, 909–909 (1968).
    [CrossRef]
  15. D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287–301 (1973).
    [CrossRef]
  16. 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]
  17. P. S. Theocaris, “Surface topography by caustics,” Appl. Opt. 15, 1629–1638 (1976).
    [CrossRef]
  18. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
    [CrossRef]
  19. P. S. Theocaris, “Properties of caustics from conic reflectors. 1: Meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
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    [CrossRef]
  23. G. L. Strobel, and D. L. Shealy, “Caustic surface analysis for a gradient-index lens,” J. Opt. Soc. Am. 70, 1264–1269 (1980).
    [CrossRef]
  24. P. S. Theocaris, and J. G. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophe theory,” Appl. Opt. 21, 1080–1091 (1982).
    [CrossRef]
  25. D. G. Burkhard, and D. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt. 21, 3299–3306 (1982).
    [CrossRef]
  26. P. S. Theocaris and T. P. Philippidis, “Possibilities of reflected caustics due to an improved optical arrangement: some further aspects,” Appl. Opt. 23, 3667–3675 (1984).
    [CrossRef]
  27. M. V. Berry, “Disruption of images: the caustic-touching theorem,”J. Opt. Soc. Am. A 4, 561–569 (1987).
    [CrossRef]
  28. A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 27, 516–522 (1988).
    [CrossRef]
  29. P. S. Theocaris, “Multicusp caustics formed from reflections of warped surfaces,” Appl. Opt. 27, 780–789 (1988).
    [CrossRef]
  30. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
    [CrossRef]
  31. I. H. Al-Ahdali and D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
    [CrossRef]
  32. D. R. J. Chillingworth, G. R. Danesh-Naroule, and B. S. Westcott, “On ray-tracing via caustic geometry,” IEEE Trans. Antennas Propag. 38, 625–632 (1990).
    [CrossRef]
  33. M. R. Hatch and D. E. Stoltzmann, “Extending the caustic test to general aspheric surfaces,” Appl. Opt. 31, 4343–4349 (1992).
    [CrossRef]
  34. D. P. K. Banerjee, R. V. Willstrop, and B. G. Anandarao, “Improving the accuracy of the caustic test,” Appl. Opt. 37, 1227–1230 (1998).
    [CrossRef]
  35. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
    [CrossRef]
  36. G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
    [CrossRef]
  37. P. Arguijo and M. Strojnik Scholl, “Exact ray-trace beam for an off-axis paraboloid surface,” Appl. Opt. 42, 3284–3289 (2003).
    [CrossRef]
  38. J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080–6089 (2004).
    [CrossRef]
  39. P. Su, J. Hudman, J. Sasian, and W. Dallas, “Dual beam generation at a ray caustic by a multiplexing computer-generated hologram,” Opt. Express 13, 4843–4847 (2005).
    [CrossRef]
  40. M. Avendaño-Alejo and R. Díaz-Uribe, “Testing a fast off-axis parabolic mirror by using tilted null screens,” Appl. Opt. 45, 2607–2614 (2006).
    [CrossRef]
  41. G. Essl, “Computation of wave fronts on a disk I: numerical experiments,” Electr. Notes Theor. Comput. Sci. 161, 25–41 (2006).
  42. M. Avendaño-Alejo, R. Díaz-Uribe, and I. Moreno, “Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal,” J. Opt. Soc. Am. A 25, 1586–1593 (2008).
    [CrossRef]
  43. E. Román-Hernández and G. Silva-Ortigoza, “Exact computation of image disruption under reflection on a smooth surface and Ronchigrams,” Appl. Opt. 47, 5500–5518 (2008).
    [CrossRef]
  44. M. Faryad, “High frequency expressions for the field in the caustic region of a cylindrical reflector placed in chiral medium,” Prog. Electromagn. Res. 76, 153–182 (2007).
    [CrossRef]
  45. M. Avendaño-Alejo, V. I. Moreno-Oliva, M. Campos-García, and R. Díaz-Uribe, “Quantitative evaluation of an off-axis parabolic mirror by using a tilted null screen,” Appl. Opt. 48, 1008–1015 (2009).
    [CrossRef]
  46. E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, R. Silva-Ortigoza, and J. Velázquez-Castro, “Describing the structure of ronchigrams when the grating is placed at the caustic region: the parabolical mirror,” J. Opt. Soc. Am. A 27, 832–845 (2010).
    [CrossRef]
  47. M. Avendaño-Alejo, L. Castañeda, and I. Moreno, “Properties of caustics produced by a positive lens: meridional rays,” J. Opt. Soc. Am. A 27, 2252–2260 (2010).
    [CrossRef]
  48. M. Avendañ-Alejo, D. González-Utrera, and L. Castañeda, “Caustics in a meridional plane produced by plano-convex conic lenses,” J. Opt. Soc. Am. A 28, 2619–2628 (2011).
    [CrossRef]
  49. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437 (1964).
    [CrossRef]
  50. J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge University, 1992).
  51. V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).
  52. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable MapsI (Birkhäuser, 1995).
  53. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).
  54. J. A. Hoffnagle and D. L. Shealy, “Stavroudis’s solution to the eikonal equation for multielement optical systems,” J. Opt. Soc. Am. A 28, 1312–1321 (2011).
    [CrossRef]

2011 (3)

2010 (2)

2009 (2)

2008 (3)

2007 (1)

M. Faryad, “High frequency expressions for the field in the caustic region of a cylindrical reflector placed in chiral medium,” Prog. Electromagn. Res. 76, 153–182 (2007).
[CrossRef]

2006 (2)

G. Essl, “Computation of wave fronts on a disk I: numerical experiments,” Electr. Notes Theor. Comput. Sci. 161, 25–41 (2006).

M. Avendaño-Alejo and R. Díaz-Uribe, “Testing a fast off-axis parabolic mirror by using tilted null screens,” Appl. Opt. 45, 2607–2614 (2006).
[CrossRef]

2005 (1)

2004 (1)

2003 (1)

2002 (1)

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
[CrossRef]

2001 (1)

1998 (1)

1995 (2)

1992 (1)

1990 (2)

I. H. Al-Ahdali and D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
[CrossRef]

D. R. J. Chillingworth, G. R. Danesh-Naroule, and B. S. Westcott, “On ray-tracing via caustic geometry,” IEEE Trans. Antennas Propag. 38, 625–632 (1990).
[CrossRef]

1989 (1)

1988 (2)

1987 (1)

1984 (1)

1982 (2)

1980 (1)

1979 (1)

1978 (2)

1977 (2)

1976 (3)

1973 (2)

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]

D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

1968 (3)

I. Lazar and L. A. DeAcetis, “Rays from a parabolic reflector for an off focal-points source,” Am. J. Phys. 36, 139–141 (1968).
[CrossRef]

R. B. Harris, “Rays from a parabolic reflector for an off focal-point source,” Am. J. Phys. 36, 1022–1023 (1968).
[CrossRef]

L. A. DeAcetis, “Wavefronts from a parabolic reflector for an off-focal source and comment on the off-axis caustic,” Am. J. Phys. 36, 909–909 (1968).
[CrossRef]

1965 (1)

1964 (2)

Al-Ahdali, I. H.

Anandarao, B. G.

Arguijo, P.

Arnold, V. I.

V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable MapsI (Birkhäuser, 1995).

Avendañ-Alejo, M.

Avendaño-Alejo, M.

Banerjee, D. P. K.

Berry, M. V.

Bruce, J. W.

J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge University, 1992).

Burkhard, D. G.

Campos-García, M.

Carvente-Muñoz, O.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
[CrossRef]

Castañeda, L.

Castro-Ramos, J.

Chang, R. S.

Chillingworth, D. R. J.

D. R. J. Chillingworth, G. R. Danesh-Naroule, and B. S. Westcott, “On ray-tracing via caustic geometry,” IEEE Trans. Antennas Propag. 38, 625–632 (1990).
[CrossRef]

Cordero-Dávila, A.

Cornejo, A.

Dallas, W.

Danesh-Naroule, G. R.

D. R. J. Chillingworth, G. R. Danesh-Naroule, and B. S. Westcott, “On ray-tracing via caustic geometry,” IEEE Trans. Antennas Propag. 38, 625–632 (1990).
[CrossRef]

de Ita Prieto, O.

DeAcetis, L. A.

I. Lazar and L. A. DeAcetis, “Rays from a parabolic reflector for an off focal-points source,” Am. J. Phys. 36, 139–141 (1968).
[CrossRef]

L. A. DeAcetis, “Wavefronts from a parabolic reflector for an off-focal source and comment on the off-axis caustic,” Am. J. Phys. 36, 909–909 (1968).
[CrossRef]

Díaz-Uribe, R.

Essl, G.

G. Essl, “Computation of wave fronts on a disk I: numerical experiments,” Electr. Notes Theor. Comput. Sci. 161, 25–41 (2006).

Faryad, M.

M. Faryad, “High frequency expressions for the field in the caustic region of a cylindrical reflector placed in chiral medium,” Prog. Electromagn. Res. 76, 153–182 (2007).
[CrossRef]

Fronczek, R. C.

Gdoutos, E. E.

Giblin, P. J.

J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge University, 1992).

González-Utrera, D.

Gusein-Zade, S. M.

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable MapsI (Birkhäuser, 1995).

Harris, R. B.

R. B. Harris, “Rays from a parabolic reflector for an off focal-point source,” Am. J. Phys. 36, 1022–1023 (1968).
[CrossRef]

Hatch, M. R.

Henry, J.

Hoffnagle, J. A.

Howard, J. E.

Hudman, J.

Kassim, A. M.

Lazar, I.

I. Lazar and L. A. DeAcetis, “Rays from a parabolic reflector for an off focal-points source,” Am. J. Phys. 36, 139–141 (1968).
[CrossRef]

Malacara, D.

Marciano-Melchor, M.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
[CrossRef]

Michopoulos, J. G.

Moreno, I.

Moreno-Oliva, V. I.

Philippidis, T. P.

Roman-Hernández, E.

E. Romań-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 12, 055705 (2011).

E. Romań-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustic of a spherical wave reflected by an arbitrary smooth surface,” J. Opt. Soc. Am. A 26, 2295–2305 (2009).
[CrossRef]

Román-Hernández, E.

Ronchi, V.

Santiago-Santiago, J. G.

Sasian, J.

Scarborough, J. B.

Scholl, M. Strojnik

Shealy, D.

Shealy, D. L.

J. A. Hoffnagle and D. L. Shealy, “Stavroudis’s solution to the eikonal equation for multielement optical systems,” J. Opt. Soc. Am. A 28, 1312–1321 (2011).
[CrossRef]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[CrossRef]

I. H. Al-Ahdali and D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
[CrossRef]

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef]

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 27, 516–522 (1988).
[CrossRef]

G. L. Strobel, and D. L. Shealy, “Caustic surface analysis for a gradient-index lens,” J. Opt. Soc. Am. 70, 1264–1269 (1980).
[CrossRef]

D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
[CrossRef]

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]

D. L. Shealy and D. G. Burkhard, “Flux density ray propagation in discrete index media expressed in terms of the intrinsic geometry of the reflecting surface,” Opt. Acta 20, 287–301 (1973).
[CrossRef]

Silva-Ortigoza, G.

E. Romań-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 12, 055705 (2011).

E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, R. Silva-Ortigoza, and J. Velázquez-Castro, “Describing the structure of ronchigrams when the grating is placed at the caustic region: the parabolical mirror,” J. Opt. Soc. Am. A 27, 832–845 (2010).
[CrossRef]

E. Romań-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustic of a spherical wave reflected by an arbitrary smooth surface,” J. Opt. Soc. Am. A 26, 2295–2305 (2009).
[CrossRef]

E. Román-Hernández and G. Silva-Ortigoza, “Exact computation of image disruption under reflection on a smooth surface and Ronchigrams,” Appl. Opt. 47, 5500–5518 (2008).
[CrossRef]

J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080–6089 (2004).
[CrossRef]

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
[CrossRef]

G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
[CrossRef]

Silva-Ortigoza, R.

Silva-Ortigoza, Ramón

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and Ramón Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
[CrossRef]

Stalzer,

Stavroudis, O. N.

Stoltzmann, D. E.

Strobel, G. L.

Su, P.

Theocaris, P. S.

Varchenko, A. N.

V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable MapsI (Birkhäuser, 1995).

Velázquez-Castro, J.

Westcott, B. S.

D. R. J. Chillingworth, G. R. Danesh-Naroule, and B. S. Westcott, “On ray-tracing via caustic geometry,” IEEE Trans. Antennas Propag. 38, 625–632 (1990).
[CrossRef]

Willstrop, R. V.

Am. J. Phys. (3)

I. Lazar and L. A. DeAcetis, “Rays from a parabolic reflector for an off focal-points source,” Am. J. Phys. 36, 139–141 (1968).
[CrossRef]

R. B. Harris, “Rays from a parabolic reflector for an off focal-point source,” Am. J. Phys. 36, 1022–1023 (1968).
[CrossRef]

L. A. DeAcetis, “Wavefronts from a parabolic reflector for an off-focal source and comment on the off-axis caustic,” Am. J. Phys. 36, 909–909 (1968).
[CrossRef]

Appl. Opt. (25)

I. H. Al-Ahdali and D. L. Shealy, “Optimization of three- and four-element lens systems by minimizing the caustic surfaces,” Appl. Opt. 29, 4551–4559 (1990).
[CrossRef]

M. R. Hatch and D. E. Stoltzmann, “Extending the caustic test to general aspheric surfaces,” Appl. Opt. 31, 4343–4349 (1992).
[CrossRef]

D. P. K. Banerjee, R. V. Willstrop, and B. G. Anandarao, “Improving the accuracy of the caustic test,” Appl. Opt. 37, 1227–1230 (1998).
[CrossRef]

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437 (1964).
[CrossRef]

J. B. Scarborough, “The caustic curve of an off-axis parabola,” Appl. Opt. 3, 1445–1445 (1964).
[CrossRef]

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]

P. S. Theocaris, “Surface topography by caustics,” Appl. Opt. 15, 1629–1638 (1976).
[CrossRef]

D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
[CrossRef]

P. S. Theocaris, and E. E. Gdoutos, “Distance measuring based on caustics,” Appl. Opt. 16, 722–728 (1977).
[CrossRef]

P. S. Theocaris, “Properties of caustics from conic reflectors. 1: Meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
[CrossRef]

J. E. Howard, “Imaging properties of off-axis parabolic mirrors,” Appl. Opt. 18, 2714–2722 (1979).
[CrossRef]

P. S. Theocaris, and J. G. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophe theory,” Appl. Opt. 21, 1080–1091 (1982).
[CrossRef]

D. G. Burkhard, and D. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt. 21, 3299–3306 (1982).
[CrossRef]

P. S. Theocaris and T. P. Philippidis, “Possibilities of reflected caustics due to an improved optical arrangement: some further aspects,” Appl. Opt. 23, 3667–3675 (1984).
[CrossRef]

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 27, 516–522 (1988).
[CrossRef]

P. S. Theocaris, “Multicusp caustics formed from reflections of warped surfaces,” Appl. Opt. 27, 780–789 (1988).
[CrossRef]

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
[CrossRef]

G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028 (2001).
[CrossRef]

P. Arguijo and M. Strojnik Scholl, “Exact ray-trace beam for an off-axis paraboloid surface,” Appl. Opt. 42, 3284–3289 (2003).
[CrossRef]

J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt. 43, 6080–6089 (2004).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Schematic drawing of the optical system and the vectors used to compute the optical path length from some point in the object space through the optical system to the point in image space whose coordinates are X, Y, and Z. In this diagram, s⃗=(s1,s2,s3) denotes the position of the point light source, I^ the direction of an emitted light ray, r⃗=(x,y,f(x,y)) the point on the smooth surface where the emitted light ray is refracted in the direction R^, and N^ the normal vector to the smooth surface at the point of refraction.

Fig. 2.
Fig. 2.

(a) Here we present the refracting surface, some refracted light rays, and the caustic associated with the refraction of a spherical wavefront. (b) Some light rays at the caustic region.

Fig. 3.
Fig. 3.

(a) The evolution of the refracted wavefront and the caustic associated with the refraction of a spherical wavefront. (b) Some wavefronts at the caustic region.

Fig. 4.
Fig. 4.

(a) Here we present the refracting surface, some refracted light rays, and the caustic associated with the refraction of a plane wavefront. (b) Some right rays at the caustic region.

Fig. 5.
Fig. 5.

(a) The evolution of the refracted plane wavefront and the caustic. (b) Some wavefronts at the caustic region.

Fig. 6.
Fig. 6.

Branch of the caustic given by Eq. (26) with the plus sign for a parabolic refracting surface and different positions of the point light source: (a) (0, 0, 300mm), (b) (5 mm, 5 mm, 300mm), (c) (10 mm, 10 mm, 300mm), (d) (15 mm, 15 mm, 300mm), (e) (20 mm, 20 mm, 300mm), and (f) (25 mm, 25 mm, 300mm).

Fig. 7.
Fig. 7.

Branch of the caustic given by Eq. (26) with the minus sign for a parabolic refracting surface and different positions of the point light source: (a) (0, 0, 300mm), (b) (5 mm, 5 mm, 300mm), (c) (10 mm, 10 mm, 300mm), (d) (15 mm, 15 mm, 300mm), (e) (20 mm, 20 mm, 300mm), and (f) (25 mm, 25 mm, 300mm).

Fig. 8.
Fig. 8.

Here we present the intersections of the caustic shown in Figs. 6 and 7 with the plane Z=700mm.

Fig. 9.
Fig. 9.

Refracted wavefronts before the caustic C=500mm, at the caustic C=800mm, 900 mm, 1100 mm, 1400 mm and after de caustic C=2000mm for a parabolic refracting surface and the point light source at (5 mm, 5 mm, 300mm).

Fig. 10.
Fig. 10.

Here we show the refracting parabolic surface, the caustic, and some refracted wavefronts when the point light source is placed at (5 mm, 5 mm, 300mm).

Equations (40)

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X⃗=r⃗+lR^,
R^=γI^+ΩN^,
γn1n2,
Ωγ(I^N^)1γ2[1(I^N^)2].
I^=r⃗s⃗|r⃗s⃗|=(xs1,ys2,fs3)(xs1)2+(ys2)2+(fs3)2.
G(x,y,z)=f(x,y)z.
N⃗=∇⃗G=(fx,fy,1),
N^=∇⃗G|∇⃗G|=(fx,fy,1)1+fx2+fy2.
R^=h⃗α,
h⃗=γ|∇⃗G|2(r⃗s⃗)[γ(r⃗s⃗)·∇⃗G+Ψ]∇⃗G,α=|r⃗s⃗||∇⃗G|2,Ψ=(1γ2)|r⃗s⃗|2|∇⃗G|2+γ2[(r⃗s⃗)·∇⃗G]2.
X⃗=r⃗+l(h⃗α).
Φ=n1(r⃗s⃗)·I^+n2(X⃗r⃗)·R^.
Φ(X⃗,s⃗,x,y)=n2X⃗·R^+k,
k(s⃗,x,y)n1|r⃗s⃗|n2(r⃗·R^).
n2X⃗·R^+k=n2C,
n2X⃗·R^x+kx=0,
n2X⃗·R^y+ky=0,
X⃗(s⃗,x,y,C)=r⃗(x,y)+[Cγ|r⃗(x,y)s⃗|]R^(s⃗,x,y).
l=Cγ|r⃗(x,y)s⃗|.
x=x(X,Y,Z),y=y(X,Y,Z).
Φ˜(X,Y,Z)n2X⃗·R^(x(X,Y,Z),y(X,Y,Z))+k(x(X,Y,Z),y(X,Y,Z)).
J(x,y,C)=det((X,Y,Z)(x,y,C))=(X⃗x)·(X⃗y×X⃗C)=0.
H2(Cγ|r⃗(x,y)s⃗|α)2+H1(Cγ|r⃗(x,y)s⃗|α)+H0=0,
H2(x,y)=h⃗·[(h⃗x)×(h⃗y)],H1(x,y)=h⃗·[(r⃗x)×(h⃗y)+(h⃗x)×(r⃗y)],H0(x,y)=h⃗·[(r⃗x)×(r⃗y)].
C=C±(x,y)γ|r⃗(x,y)s⃗|+α(H1±H124H2H02H2).
X⃗=X⃗c±r⃗+(H1±H124H2H02H2)h⃗.
C(x,y)γ|r⃗s⃗|α(H0H1).
X⃗=X⃗c=r⃗(H0H1)h⃗.
Φ=n1(r⃗r⃗0)·z^+n2(X⃗r⃗)·R^.
z=f(x,y)=12a(x2+y2),
k(s⃗,x,y)=12a{n1Ωn2[Δa(x2+y2)(2s3(x2+y2)γ+ϒ)](a2+x2+y2)Ω},
Ω=4a2[s32+(s1x)2+(s2y)2]4as3(x2+y2)+(x2+y2)2,Δ=4a3s3(x2+y2)γ2a2((s12x)x+(s22y)y)(x2+y2)γ+(x2+y2)3γ+4a4(s1x+x2s2y+y2)γ,ϒ=4a3s3(x2+y2)+8a3s3[s1x+x2s2y+y2]γ2+4as3(x2+y2)2(1+γ2)(x2+y2)3(1+γ2)+4a4[s32+(s1x)2+(s2y)2((s1x)2+(s2y)2)γ2]+a2[(x2+y2)[4s12+4s22+4s328s1x8s2y+5(x2+y2)]4[x2(s22+s32+x(s1+x))s2x(2s1+x)y+(s12+s32s1x+2x2)y2s2y3+y4]γ2].
X=x+{[2a2(s1x)+2as3xx3+2s1y2xy(2s2+y)]γ+xϒa}Σ,Y=y+{[2a2(s2y)+2as3yy3+2s2x2xy(2s1+x)]γ+yϒa}Σ,Z=f{[(x2+y2)2a2a(s1xx2+s2yy2)2s3(x2+y2)]γ+ϒ}Σ,
Σ=γΩ2aτ2(a2+x2+y2)Ω.
r⃗x=(1,0,x/a),r⃗y=(1,0,y/a),h1=12a3[2a2s3xγ+2a3(s1+x)γ+a(x32s1y2+xy(2s2+y))γxϒ],h2=12a3[2a2s3yγ+2a3(s2+y)γ+a(y32s2x2+xy(2s1+x))γyϒ],h3=12a3[(x2+y2)2γ+2a2(s1x+x2s2y+y2)γ+a[2s3(x2+y2)γ+ϒ].
k=(x2+y2)[an1n2a2+(x2+y2)(1γ2)]2(a2+x2+y2).
X=xx[aγa2(x2+y2)(1+γ2)][(x2+y2)γ2aτ]2a(a2+x2+y2),Y=yy[aγa2(x2+y2)(1+γ2)][(x2+y2)γ2aτ]2a(a2+x2+y2),Z=c2(x2+y2)[(x2+y2)γ+aa2(x2+y2)(1+γ2)][(x2+y2)γ2aτ]2a(a2+x2+y2).
C+=(a2+x2+y2)[a2+(x2+y2)(1γ2)][aγ+a2+(x2+y2)(1γ2)]a2[(x2+y2)(1γ2)+a2(1+γ2)2aγa2+(x2+y2)(1γ2)],C=aγ+a2+(x2+y2)(1γ2)1γ2,
X+=x(x2+y2)(1+γ2)a2,Y+=y(x2+y2)(1+γ2)a2,Z+=a[2a2+3(x2+y2)(1γ2)]+2γ[a2+(x2+y2)(1γ2)]3/22a2(1γ2),
X⃗={0,0,(x2+y2)2a+a+γa2+(x2+y2)(1γ2)1γ2}.

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