Abstract

In this work we use the geometrical point of view of the Ronchi test and the caustic-touching theorem to describe the structure of the ronchigrams for a parabolical mirror when the point light source is on and off the optical axis and the grating is placed at the caustic associated with the reflected light rays. We find that for a given position of the point light source the structure of the ronchigram is determined by the form of the caustic and the relative position between the grating and the caustic. We remark that the closed loop fringes commonly observed in the ronchigrams appear when the grating and the caustic are tangent to each other. Furthermore, we find that the caustic locally has singularities of the purse or hyperbolic umbilic type, and the ronchigram obtained when the grating is located at certain specific positions at the caustic locally is of the serpentine type. The main motivation of this work is that nowadays a quantitative analysis of the Ronchi test is applied only when the grating is outside the caustic, and we claim that by working at the caustic, the sensitivity of the Ronchi test will be improved. Therefore, a clear understanding of the properties of the ronchigrams when the grating is placed at the caustic will be needed to extend the Ronchi test to that region.

© 2010 Optical Society of America

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References

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  1. D. Malacara, A. Cornejo, and M. V. R. K. Murty, “Bibliography of various optical testing methods,” Appl. Opt.  14, 1065–1065 (1975).
    [CrossRef] [PubMed]
  2. A. Cornejo, H. J. Caulfied, and W. Friday, “Testing of optical surfaces: A bibliography,” Appl. Opt.  20, 4148–4148 (1981).
    [CrossRef]
  3. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt.  3, 437–451 (1964).
    [CrossRef]
  4. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1978), Chap. 9.
  5. M. Mansuripur, “The Ronchi test,” Opt. Photonics News , 42–46 (1997).
  6. D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt.  4, 1371–1374 (1965).
    [CrossRef]
  7. A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assc.  68, 180–191 (1958).
  8. A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. New South Wales  43, 19 (1960).
  9. D. Malacara and A. Cornejo, “Null Ronchi test for aspherical surfaces,” Appl. Opt.  13, 1778–1780 (1974).
    [CrossRef] [PubMed]
  10. T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt.  23, 3676–3679 (1984).
    [CrossRef] [PubMed]
  11. A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Null Hartmann and Ronchi-Hartmann tests,” Appl. Opt.  29, 4618–4621 (1990).
    [CrossRef] [PubMed]
  12. A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt.  31, 2370–2376 (1992).
    [CrossRef] [PubMed]
  13. A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt.  41, 3866–3873 (2002).
    [CrossRef] [PubMed]
  14. 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] [PubMed]
  15. M. V. Berry, “Disruption of images: the caustic-touching theorem,” J. Opt. Soc. Am. A  4, 561–569 (1987).
    [CrossRef]
  16. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am.  66, 795–800 (1976).
    [CrossRef]
  17. 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] [PubMed]
  18. 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]
  19. E. Román-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  20, 2295–2305 (2009).
    [CrossRef]
  20. 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]
  21. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt.  20, 897–909 (1981).
    [CrossRef] [PubMed]
  22. P. S. Theocaris, “Surface topography by caustics,” Appl. Opt.  15, 1629–1638 (1976).
    [CrossRef] [PubMed]
  23. P. S. Theocaris, “Properties of caustics from conic reflectors. 1: Meridional rays,” Appl. Opt.  16, 1705–1716 (1977).
    [CrossRef] [PubMed]
  24. 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]
  25. 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, Pure Appl. Opt.  4, 358–365 (2002).
    [CrossRef]
  26. J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt.  43, 6080–6088 (2004).
    [CrossRef] [PubMed]
  27. 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]
  28. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt.  15, 2588–2596 (1976).
    [CrossRef] [PubMed]
  29. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt.  28, 601606 (1989).
    [CrossRef]
  30. 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 (1989).
    [CrossRef]
  31. D. G. Burkhard and D. L. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt.  21, 32993306 (1982).
    [CrossRef]

2009 (1)

E. Román-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  20, 2295–2305 (2009).
[CrossRef]

2008 (2)

2004 (1)

2002 (2)

A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt.  41, 3866–3873 (2002).
[CrossRef] [PubMed]

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, Pure Appl. Opt.  4, 358–365 (2002).
[CrossRef]

2001 (1)

1997 (1)

M. Mansuripur, “The Ronchi test,” Opt. Photonics News , 42–46 (1997).

1992 (1)

1990 (1)

1989 (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 (1989).
[CrossRef]

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt.  28, 601606 (1989).
[CrossRef]

1988 (1)

1987 (1)

1984 (1)

1982 (1)

D. G. Burkhard and D. L. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt.  21, 32993306 (1982).
[CrossRef]

1981 (2)

1977 (1)

1976 (3)

1975 (1)

1974 (1)

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] [PubMed]

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]

1965 (1)

1964 (1)

1960 (1)

A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. New South Wales  43, 19 (1960).

1958 (1)

A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assc.  68, 180–191 (1958).

Al-Ahdali, I. H.

Berry, M. V.

Burkhard, D. G.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt.  28, 601606 (1989).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt.  21, 32993306 (1982).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt.  20, 897–909 (1981).
[CrossRef] [PubMed]

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]

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]

Cabrera-Peláez, V.

Cardona-Nuñez, O.

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, Pure Appl. Opt.  4, 358–365 (2002).
[CrossRef]

Castro-Ramos, J.

Caulfied, H. J.

Cordero-Dávila, A.

Cornejo, A.

Cornejo-Rodriguez, A.

de Ita Prieto, O.

Díaz-Anzures, J.

Friday, W.

Fronczek, R. C.

Hoffnagle, J. A.

Kassim, A. M.

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt.  28, 601606 (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] [PubMed]

Malacara, D.

Mansuripur, M.

M. Mansuripur, “The Ronchi test,” Opt. Photonics News , 42–46 (1997).

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, Pure Appl. Opt.  4, 358–365 (2002).
[CrossRef]

Murty, M. V. R. K.

Román-Hernández, E.

E. Román-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  20, 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] [PubMed]

Ronchi, V.

Santiago-Santiago, J. G.

E. Román-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  20, 2295–2305 (2009).
[CrossRef]

Shealy, D. L.

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]

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt.  28, 601606 (1989).
[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 (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] [PubMed]

D. G. Burkhard and D. L. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt.  21, 32993306 (1982).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt.  20, 897–909 (1981).
[CrossRef] [PubMed]

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

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]

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]

Sherwood, A. A.

A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. New South Wales  43, 19 (1960).

A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assc.  68, 180–191 (1958).

Silva-Ortigoza, G.

E. Román-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  20, 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] [PubMed]

J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt.  43, 6080–6088 (2004).
[CrossRef] [PubMed]

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, Pure Appl. Opt.  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.

E. Román-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  20, 2295–2305 (2009).
[CrossRef]

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, Pure Appl. Opt.  4, 358–365 (2002).
[CrossRef]

Stavroudis, O. N.

Theocaris, P. S.

Yatagai, T.

Appl. Opt. (21)

A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt.  28, 601606 (1989).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt.  21, 32993306 (1982).
[CrossRef]

V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt.  3, 437–451 (1964).
[CrossRef]

D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt.  4, 1371–1374 (1965).
[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] [PubMed]

D. Malacara and A. Cornejo, “Null Ronchi test for aspherical surfaces,” Appl. Opt.  13, 1778–1780 (1974).
[CrossRef] [PubMed]

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “Bibliography of various optical testing methods,” Appl. Opt.  14, 1065–1065 (1975).
[CrossRef] [PubMed]

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

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

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

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt.  20, 897–909 (1981).
[CrossRef] [PubMed]

A. Cornejo, H. J. Caulfied, and W. Friday, “Testing of optical surfaces: A bibliography,” Appl. Opt.  20, 4148–4148 (1981).
[CrossRef]

T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt.  23, 3676–3679 (1984).
[CrossRef] [PubMed]

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] [PubMed]

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 (1989).
[CrossRef]

A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Null Hartmann and Ronchi-Hartmann tests,” Appl. Opt.  29, 4618–4621 (1990).
[CrossRef] [PubMed]

A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Ronchi and Hartmann tests with the same mathematical theory,” Appl. Opt.  31, 2370–2376 (1992).
[CrossRef] [PubMed]

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]

A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt.  41, 3866–3873 (2002).
[CrossRef] [PubMed]

J. Castro-Ramos, O. de Ita Prieto, and G. Silva-Ortigoza, “Computation of the disk of least confusion for conic mirrors,” Appl. Opt.  43, 6080–6088 (2004).
[CrossRef] [PubMed]

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] [PubMed]

J. Br. Astron. Assc. (1)

A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assc.  68, 180–191 (1958).

J. Opt. A, Pure Appl. Opt. (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, Pure Appl. Opt.  4, 358–365 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

M. V. Berry, “Disruption of images: the caustic-touching theorem,” J. Opt. Soc. Am. A  4, 561–569 (1987).
[CrossRef]

E. Román-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  20, 2295–2305 (2009).
[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]

J. Proc. R. Soc. New South Wales (1)

A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. New South Wales  43, 19 (1960).

Opt. Acta (1)

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]

Opt. Photonics News (1)

M. Mansuripur, “The Ronchi test,” Opt. Photonics News , 42–46 (1997).

Other (1)

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1978), Chap. 9.

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Figures (16)

Fig. 1
Fig. 1

Branch of the caustic given by Eqs. (17) with the plus sign for different positions of the point light source.

Fig. 2
Fig. 2

Branch of the caustic given by Eqs. (17) with the minus sign for different positions of the point light source.

Fig. 3
Fig. 3

Intersections of the caustic shown in Figs. 1, 2 with the plane z = 710 mm . From these plots it is clear that the caustic locally has singularities of the purse or hyperbolic umbilic type.

Fig. 4
Fig. 4

(a) Object space: Ronchi ruling and the caustic, which is a circle of radius R c = 18.28 mm , with its center. (b) Image space: The corresponding shadow or Ronchigram.

Fig. 5
Fig. 5

(a)–(c) Object space and (d)–(f) image space for n d = 20 , 10 , 0 , respectively.

Fig. 6
Fig. 6

(a) Object space: Ronchi ruling and the caustic. (b) Image space: the associated Ronchigram. For Θ = 0 and s 1 = 500 mm .

Fig. 7
Fig. 7

(a)–(c) Object space and (d)–(f) image space for n d = 22 , 20 , 18 , respectively. For Θ = 0 and s 1 = 500 mm .

Fig. 8
Fig. 8

(a)–(c) Object space and (d)–(f) image space for n d = 2 , 0 , 2 , respectively. For Θ = 0 and s 1 = 500 mm .

Fig. 9
Fig. 9

(a)–(c) Object space and (d)–(f) image space for n d = 18 , 20 , 22 , respectively. For Θ = 0 and s 1 = 500 mm .

Fig. 10
Fig. 10

(a) Object space: Ronchi ruling and the caustic. (b) Image space: the associated Ronchigram. For Θ = π 4 and s 1 = 500 mm .

Fig. 11
Fig. 11

(a)–(c) Object space and (d)–(f) image space for n d = 240 , 227.4 , 225 , respectively. For Θ = π 4 and s 1 = 500 mm .

Fig. 12
Fig. 12

(a)–(c) Object space and (d)–(f) image space for n d = 204 , 201.1 , 199 , respectively. For Θ = π 4 and s 1 = 500 mm .

Fig. 13
Fig. 13

(a)–(c) Object space and (d)–(f) image space for n d = 178.5 , 176.5 , 175 , respectively. For Θ = π 4 and s 1 = 500 mm .

Fig. 14
Fig. 14

Simulation of the Ronchi ruling and the caustic curve.

Fig. 15
Fig. 15

Shadows of the Ronchi rulings or ronchigrams. For Θ = π 4 and s 1 = 500 mm .

Fig. 16
Fig. 16

(a) Object space: rulings and the caustic. (b) Image space: the associated shadows. For Θ = π 4 and s 1 = 500 mm . Observe that for these rulings the shadow is the well-known serpentine pattern.

Equations (31)

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T 1 ( x , y , z 0 ) = x + [ z 0 f ( x , y ) ] ( h 1 ( x , y , s ) h 3 ( x , y , s ) ) ,
T 2 ( x , y , z 0 ) = y + [ z 0 f ( x , y ) ] ( h 2 ( x , y , s ) h 3 ( x , y , s ) ) ,
h 1 = ( x s 1 ) ( 1 f x 2 + f y 2 ) 2 f x [ f y ( y s 2 ) + s 3 f ] ,
h 2 = ( y s 2 ) ( 1 + f x 2 f y 2 ) 2 f y [ f x ( x s 1 ) + s 3 f ] ,
h 3 = ( f s 3 ) ( 1 + f x 2 + f y 2 ) + 2 [ f x ( x s 1 ) + f y ( y s 2 ) ] ,
T c ± = r + ( H 1 ± H 1 2 4 H 2 H 0 2 H 2 ) h ,
H 0 = h [ ( r x ) × ( r y ) ] ,
H 1 = h [ ( r x ) × ( h y ) + ( h x ) × ( r y ) ] ,
H 2 = h [ ( h x ) × ( h y ) ] .
T x = Γ ( σ ) , T y = Σ ( σ ) ,
T x ( σ ) = x + [ z 0 f ( x , y ) ] ( h 1 ( x , y , s ) h 3 ( x , y , s ) ) ,
T y ( σ ) = y + [ z 0 f ( x , y ) ] ( h 2 ( x , y , s ) h 3 ( x , y , s ) ) ,
T y = T x tan Θ + n d ,
y + [ z 0 f ( x , y ) ] ( h 2 ( x , y , s ) h 3 ( x , y , s ) ) = { x + [ z 0 f ( x , y ) ] ( h 1 ( x , y , s ) h 3 ( x , y , s ) ) } tan Θ + n d .
f = a + 1 2 a ( x 2 + y 2 ) ,
r = ( x , y , a + 1 2 a ( x 2 + y 2 ) ) ,
h 1 = 1 a 2 [ 2 a s 3 x a 2 ( s 1 + x ) + 2 s 2 x y + s 1 ( x 2 y 2 ) ] ,
h 2 = 1 a 2 [ 2 a s 3 y a 2 ( s 2 + y ) + 2 s 1 x y s 2 ( x 2 y 2 ) ] ,
h 3 = 1 2 a 3 [ 2 a 4 + 2 a 3 s 3 2 a s 3 ( x 2 + y 2 ) + ( x 2 + y 2 ) 2 + a 2 ( x 2 + y 2 4 s 1 x 4 s 2 y ) ] .
T 1 ( x , y , z 0 ) = x + [ 2 a ( z 0 + a ) x 2 y 2 ] [ 2 a s 3 x a 2 ( s 1 + x ) + 2 s 2 x y + s 1 ( x 2 y 2 ) ] 2 a 4 + 2 a 3 s 3 2 a s 3 ( x 2 + y 2 ) + ( x 2 + y 2 ) 2 + a 2 ( x 2 + y 2 4 s 1 x 4 s 2 y ) ,
T 2 ( x , y , z 0 ) = y + [ 2 a ( z 0 + a ) x 2 y 2 ] [ 2 a s 3 y a 2 ( s 2 + y ) + 2 s 1 x y s 2 ( x 2 y 2 ) ] 2 a 4 + 2 a 3 s 3 2 a s 3 ( x 2 + y 2 ) + ( x 2 + y 2 ) 2 + a 2 ( x 2 + y 2 4 s 1 x 4 s 2 y ) .
H 0 = 1 2 a 3 ( a 2 + x 2 + y 2 ) ( 2 a 2 + 2 a s 3 2 s 1 x + x 2 2 s 2 y + y 2 ) ,
H 1 = 1 a 5 [ ( a 2 + x 2 + y 2 ) ( 2 a 4 + 6 a 3 s 3 + a 2 ( 2 s 1 2 + 2 s 2 2 + 4 s 3 2 6 s 1 x x 2 6 s 2 y y 2 ) + 2 ( s 1 2 + s 2 2 ) ( x 2 + y 2 ) 2 a s 3 ( 2 s 1 x + x 2 + 2 s 2 y + y 2 ) ) ] ,
H 2 = 1 2 a 7 { ( a 2 + x 2 + y 2 ) [ 2 a 6 + 10 a 5 s 3 + 16 a s 3 ( s 1 x + s 2 y ) ( x 2 + y 2 ) 4 ( s 1 2 + s 2 2 ) ( x 2 + y 2 ) 2 + 4 a 2 [ x ( 2 s 1 ( s 1 2 + s 2 2 + s 3 2 ) + ( 3 s 1 2 s 2 2 3 s 3 2 ) x + 2 s 1 x 2 ) 2 s 2 ( s 1 2 + s 2 2 + s 3 2 4 s 1 x x 2 ) y ( s 1 2 3 s 2 2 + 3 s 3 2 2 s 1 x ) y 2 + 2 s 2 y 3 ] + a 4 [ 4 s 1 2 + 4 s 2 2 + 16 s 3 2 10 s 1 x 10 s 2 y 3 ( x 2 + y 2 ) ] + 4 a 3 s 3 ( 2 s 1 2 + 2 s 2 2 + 2 s 3 2 6 s 1 x 6 s 2 y 3 ( x 2 + y 2 ) ) ] } .
T 1 c ± = x + ( H 1 ± 2 H 2 a 2 ) { 2 a s 3 x a 2 ( s 1 + x ) + 2 s 2 x y + s 1 ( x 2 y 2 ) } ,
T 2 c ± = y + ( H 1 ± 2 H 2 a 2 ) { 2 a s 3 y a 2 ( s 2 + y ) + 2 s 1 x y s 2 ( x 2 y 2 ) } ,
T 3 c ± = a + x 2 + y 2 2 a + ( H 1 ± 4 H 2 a 3 ) { 2 a 4 + 2 a 3 s 3 2 a s 3 ( x 2 + y 2 ) + ( x 2 + y 2 ) 2 + a 2 ( x 2 + y 2 4 s 1 x 4 s 2 y ) } ,
Σ 2 = 1 a 10 { 4 [ ( s 1 x ) 2 + ( s 2 y ) 2 ] ( a 2 + x 2 + y 2 ) 2 [ 4 a s 3 ( s 1 x + s 2 y ) ( x 2 + y 2 ) + ( s 1 2 + s 2 2 ) ( x 2 + y 2 ) 2 + 2 a 2 [ x 2 ( s 2 2 + 2 s 3 2 s 1 ( s 1 + x ) ) s 2 x ( 4 s 1 + x ) y + ( s 1 2 s 2 2 + 2 s 3 2 s 1 x ) y 2 s 2 y 3 ] + 4 a 3 s 3 [ x ( s 1 + x ) + y ( s 2 + y ) ] + a 4 ( ( s 1 + x ) 2 + ( s 2 + y ) 2 ) ] } .
y + [ 2 a ( z 0 + a ) x 2 y 2 ] [ 2 a s 3 y a 2 ( s 2 + y ) + 2 s 1 x y s 2 ( x 2 y 2 ) ] 2 a 4 + 2 a 3 s 3 2 a s 3 ( x 2 + y 2 ) + ( x 2 + y 2 ) 2 + a 2 ( x 2 + y 2 4 s 1 x 4 s 2 y ) = [ x + [ 2 a ( z 0 + a ) x 2 y 2 ] [ 2 a s 3 x a 2 ( s 1 + x ) + 2 s 2 x y + s 1 ( x 2 y 2 ) ] 2 a 4 + 2 a 3 s 3 2 a s 3 ( x 2 + y 2 ) + ( x 2 + y 2 ) 2 + a 2 ( x 2 + y 2 4 s 1 x 4 s 2 y ) ] tan Θ + n d .
y ( 2 z 0 + a ) = x ( 2 z 0 + a ) tan Θ n d a .
y = x tan Θ n D with D d a 2 z 0 + a ,

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