Abstract

The aim of this work is threefold: first we obtain analytical expressions for the wavefront train and the caustic associated with the refraction of a plane wavefront by an axicon lens, second we describe the structure of the ronchigram when the ronchiruling is placed at the flat surface of the axicon and the screen is placed at different relative positions to the caustic region, and third we describe in detail the structure of the null ronchigrating for this system; that is, we obtain the grating such that when it is placed at the flat surface of the axicon its associated pattern, at a given plane perpendicular to the optical axis, is a set of parallel fringes. We find that the caustic has only one branch, which is a segment of a line along the optical axis; the ronchigram exhibits self-intersecting fringes when the screen is placed at the caustic region, and the null ronchigrating exhibits closed loop rulings if we want to obtain its associated pattern at the caustic region.

© 2014 Optical Society of America

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  1. V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).
  2. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhauser, 1995), Vol. I.
  3. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).
  4. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).
  5. 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]
  6. 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]
  7. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
    [CrossRef]
  8. G. L. Strobel and D. L. Shealy, “Caustic surface analysis for a gradient-index lens,” J. Opt. Soc. Am. 70, 1264–1269 (1980).
    [CrossRef]
  9. D. G. Burkhard and D. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt. 21, 3299–3306 (1982).
    [CrossRef]
  10. 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]
  11. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989).
    [CrossRef]
  12. 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]
  13. P. S. Theocaris and E. E. Gdoutos, “Distance measuring based on caustics,” Appl. Opt. 16, 722–728 (1977).
    [CrossRef]
  14. P. S. Theocaris, “Properties of caustics from conic reflectors. 1: meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
    [CrossRef]
  15. 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]
  16. P. S. Theocaris, “Multicusp caustics formed from reflections of warped surfaces,” Appl. Opt. 27, 780–789 (1988).
    [CrossRef]
  17. A. Cornejo and D. Malacara, “Caustic coordinates in Platzeck-Gaviola test of conic mirrors,” Appl. Opt. 17, 18–19 (1978).
    [CrossRef]
  18. M. R. Hatch and D. E. Stoltzmann, “Extending the caustic test to general aspheric surfaces,” Appl. Opt. 31, 4343–4349 (1992).
    [CrossRef]
  19. 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]
  20. G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. Silva-Ortigoza, “Exact computation of the caustic associated with the evolution of an aberrated wavefront,” J. Opt. A 4, 358–365 (2002).
    [CrossRef]
  21. 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]
  22. 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]
  23. 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]
  24. 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]
  25. 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]
  26. M. Avendaño-Alejo, R. Diaz-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]
  27. 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]
  28. 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]
  29. M. Avendaño-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]
  30. 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 26, 2295–2305 (2009).
    [CrossRef]
  31. 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]
  32. E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts, light rays and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 13, 055705 (2011).
    [CrossRef]
  33. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  34. J. H. McLeod, “Axicons and their uses,” J. Opt. Soc. Am. 50, 166–169 (1960).
    [CrossRef]
  35. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
    [CrossRef]
  36. J. L. Rayces, “Formation of axicon images,” J. Opt. Soc. Am. 48, 576–578 (1958).
    [CrossRef]
  37. V. Ronchi, “Forty years of history of a grating interferometer,” Appl. Opt. 3, 437–451 (1964).
    [CrossRef]
  38. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, D. Malacara, ed. (Wiley, 1978), Chap. 9.
  39. D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt. 4, 1371–1374 (1965).
    [CrossRef]
  40. A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of ray optics,” J. Br. Astron. Assoc. 68, 180–191 (1958).
  41. 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]
  42. 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]
  43. M. Marciano-Melchor, E. Navarro-Morales, E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, R. Silva-Ortigoza, and R. Suárez-Xique, “The point-characteristic function, wavefronts, and caustic of a spherical wave refracted by an arbitrary smooth surface,” J. Opt. Soc. Am. A 29, 1035–1046 (2012).
    [CrossRef]
  44. M. V. Berry, “Disruption of images: the caustic-touching theorem,” J. Opt. Soc. Am. A 4, 561–569 (1987).
    [CrossRef]
  45. M. Avendaño-Alejo, D. González-Utrera, N. Qureshi, L. Castañeda, and C. L. Ordoñes-Romero, “Null Ronchi-Hartmann test for a lens,” Opt. Express 18, 21131–21137 (2010).
    [CrossRef]

2012

2011

M. Avendaño-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]

E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts, light rays and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 13, 055705 (2011).
[CrossRef]

2010

2009

2008

2006

2005

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

2004

2002

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]

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

2001

1998

1992

1990

1989

1988

1987

1984

1982

1980

1978

1977

1976

1973

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]

1965

1964

1960

1958

J. L. Rayces, “Formation of axicon images,” J. Opt. Soc. Am. 48, 576–578 (1958).
[CrossRef]

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

1954

Al-Ahdali, I. H.

Anandarao, B. G.

Arnold, V. I.

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

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

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

Avendaño-Alejo, M.

Banerjee, D. P. K.

Berry, M. V.

Burkhard, D. G.

Cabrera-Peláez, V.

Campos-García, M.

Cardona-Nuñez, O.

Carvente-Muñoz, O.

G. Silva-Ortigoza, M. Marciano-Melchor, O. Carvente-Muñoz, and R. 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.

Cordero-Dávila, A.

Cornejo, A.

Cornejo-Rodriguez, A.

de Ita Prieto, O.

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

Díaz-Anzures, J.

Diaz-Uribe, R.

Díaz-Uribe, R.

Gdoutos, E. E.

González-Utrera, D.

Gusein-Zade, S. M.

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

Hatch, M. R.

Hoffnagle, J. A.

Kassim, A. M.

Malacara, D.

Marciano-Melchor, M.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

McLeod, J. H.

Moreno, I.

Moreno-Oliva, V. I.

Navarro-Morales, E.

Ordoñes-Romero, C. L.

Philippidis, T. P.

Qureshi, N.

Rayces, J. L.

Román-Hernández, E.

Ronchi, V.

Santiago-Santiago, J. G.

Shealy, D.

Shealy, D. L.

Sherwood, A. A.

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

Silva-Ortigoza, G.

M. Marciano-Melchor, E. Navarro-Morales, E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, R. Silva-Ortigoza, and R. Suárez-Xique, “The point-characteristic function, wavefronts, and caustic of a spherical wave refracted by an arbitrary smooth surface,” J. Opt. Soc. Am. A 29, 1035–1046 (2012).
[CrossRef]

E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts, light rays and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 13, 055705 (2011).
[CrossRef]

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. 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 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 R. 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.

Stavroudis, O. N.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).

Stoltzmann, D. E.

Strobel, G. L.

Suárez-Xique, R.

Theocaris, P. S.

Varchenko, A. N.

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

Velázquez-Castro, J.

Willstrop, R. V.

Appl. Opt.

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

A. Cornejo and D. Malacara, “Caustic coordinates in Platzeck-Gaviola test of conic mirrors,” Appl. Opt. 17, 18–19 (1978).
[CrossRef]

Contemp. Phys.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

J. Br. Astron. Assoc.

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

J. Opt.

E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts, light rays and caustic of a circular wave reflected by an arbitrary smooth curve,” J. Opt. 13, 055705 (2011).
[CrossRef]

J. Opt. A

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

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

M. Avendaño-Alejo, R. Diaz-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]

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]

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]

M. Avendaño-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]

M. Marciano-Melchor, E. Navarro-Morales, E. Román-Hernández, J. G. Santiago-Santiago, G. Silva-Ortigoza, R. Silva-Ortigoza, and R. Suárez-Xique, “The point-characteristic function, wavefronts, and caustic of a spherical wave refracted by an arbitrary smooth surface,” J. Opt. Soc. Am. A 29, 1035–1046 (2012).
[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 26, 2295–2305 (2009).
[CrossRef]

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]

Opt. Acta

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. Express

Other

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

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

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

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

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).

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Figures (13)

Fig. 1.
Fig. 1.

Schematic depiction of a plane wave incident along the z^ direction on an optical system with two refracting surfaces; one is the plane z=0, and the other is locally given by z=f(x,y). r0=(x,y,0) denotes the intersection of the incident ray with the reference plane, z=0. r=(x,y,f(x,y)) denotes the position of the point of refraction, N denotes the normal vector at the point of refraction, R gives the direction of the refracted light ray, and X, for fixed values of x and y, provides the position of a point on the refracted light ray.

Fig. 2.
Fig. 2.

Schematic depiction of the Ronchi test arrangement.

Fig. 3.
Fig. 3.

Axicon lens and its associated parameters. Observe that the apex of the axicon is given by (0, 0, acotα).

Fig. 4.
Fig. 4.

(a) Some incident light rays, the axicon lens, and some refracted light rays. (b) Some incident light rays, the axicon lens, some refracted light rays, and the caustic. (c) Some incident plane wavefronts, the axicon lens, some refracted wavefronts, and the caustic. (d) Some incident light rays and plane wavefronts, the axicon lens, some refracted light rays and wavefronts, and the caustic. To obtain this plot we take a=1cm, n=1.517, and α=π/3. Observe that a point of the caustic is given by (0, 0, acotα), which corresponds to the case x=0 and y=0.

Fig. 5.
Fig. 5.

(a) Ronchiruling placed at the flat surface of the axicon, the plane z=0. (b) Ronchigram at the plane z=1.5cm.

Fig. 6.
Fig. 6.

(a)–(c), (g)–(i) Six rulings at the plane z=0 and (d)–(f), (j)–(l) their associated shadows or fringes at the plane z=1.5cm. Here we have included the intersection of the caustic with the plane z=1.5cm, which is a point.

Fig. 7.
Fig. 7.

(a) Ronchiruling placed at the flat surface of the axicon, the plane z=0. (b) Ronchigram at the plane z=3.5cm.

Fig. 8.
Fig. 8.

(a) Intersections of the incident and refracted light rays, the ronchiruling placed at the plane z=0, the axicon, the caustic, and the screens with the plane y=0. In (b)–(f) we present the ronchigrams when the screen is placed at the planes (b) z=1cm, (c) z=1.5cm, (d) z=2cm, (e) z=2.5cm, and (f) z=3cm.

Fig. 9.
Fig. 9.

(a) Null ronchigrating placed at the flat surface of the axicon, the plane z=0. (b) Ronchigram at the plane z=1.5cm and the caustic, which is a point.

Fig. 10.
Fig. 10.

(a)–(c) Three curved rulings at the plane z=0 of the null ronchigrating and (d)–(f) their associated three fringes at the plane z=1.5cm and the caustic.

Fig. 11.
Fig. 11.

(a) Computed null ronchigrating at the flat surface of the axicon, the plane z=0. (b) Ronchigram at the plane z=3.5cm.

Fig. 12.
Fig. 12.

(a)–(c) Three curved rulings at the plane z=0 of the null ronchigrating and (d)–(f) their associated three fringes at the plane z=3.5cm.

Fig. 13.
Fig. 13.

Computed null ronchigrating when the pattern of parallel fringes is observed at the planes (d) z=1cm, (e) z=2cm, and (f) z=3cm.

Equations (25)

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X(x,y,τ)=r+[τγ|rr0|]R,
R=γI+ΩN,I=rr0|rr0|,N=(fx,fy,1)1+fx2+fy2,Ωγ(I·N)1γ2[1(I·N)2].
R=(Ωfx1+fx2+fy2)x^+(Ωfy1+fx2+fy2)y^+(γΩ1+fx2+fy2)z^
Ω=γ1+(1γ2)(fx2+fy2)1+fx2+fy2.
H2[τγ|rr0|]2+H1[τγ|rr0|]+H0=0,
H2(x,y)=R·[(Rx)×(Ry)],H1(x,y)=R·[(rx)×(Ry)+(Rx)×(ry)],H0(x,y)=R·[(rx)×(ry)].
τ=τ±=γ|rr0|+(H1±H124H2H02H2).
X=Xc±=r+(H1±H124H2H02H2)R.
τ=γ|rr0|H0H1,
X=Xc=r(H0H1)R
X(x,y,z0)=x+[z0f(x,y)](R·x^R·z^),Y(x,y,z0)=y+[z0f(x,y)](R·y^R·z^),Z(x,y,z0)=z0,
X(x,y,z0)=x+[z0f(x,y)](R·x^R·z^),Y(x,y,z0)=y+[z0f(x,y)](R·y^R·z^).
y=xtanΘ+md,
Tx(x)=X(x,xtanΘ+md,z0),Ty(x)=Y(x,xtanΘ+md,z0).
Ty=TxtanΘ+td,
y+[z0f(x,y)](R·y^R·z^)={x+[z0f(x,y)](R·x^R·z^)}tanΘ+td,
f=(ax2+y2)cotα,r=(x,y,(ax2+y2)cotα),
R=(Ωxcosαx2+y2)x^(Ωycosαx2+y2)y^+(nΩsinα)z^,
Ω=nsinα1n2cos2α.
X(x,y,τ)=x[τn(ax2+y2)cotα](Ωxcosαx2+y2),Y(x,y,τ)=y[τn(ax2+y2)cotα](Ωycosαx2+y2),Z(x,y,τ)=(ax2+y2)cotα+[τn(ax2+y2)cotα](nΩsinα).
H2(x,y)=0,H1(x,y)=Ω(Ωnsinα)cotαx2+y2,H0(x,y)=nΩcscα.
τ=(ax2+y2)ncotα+x2+y2secαΩ.
Xc=0,Yc=0,Zc=2csc(2α)[aΩcos2α+x2+y2(nsinαΩ)]Ω.
Tx(x)=(Ωcosα(acotαz0)+(γΩcscα)x2+m2d2(γΩsinα)x2+m2d2)x,Ty(x)=(Ωcosα(acotαz0)+(γΩcscα)x2+m2d2(γΩsinα)x2+m2d2)md.
y[x2+y2(nΩcscα)+(acotαz0)Ωcosα]=k[x2+y2(nΩsinα)],

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