Abstract

The aim of the present work is twofold: first we obtain analytical expressions for both the wavefronts and the caustic associated with the light rays reflected by a spherical mirror after being emitted by a point light source located at an arbitrary position in free space, and second, we describe, in detail, the structure of the ronchigrams when the grating or Ronchi ruling is placed at different relative positions to the caustic region and the point light source is located on and off the optical axis. We find that, in general, the caustic has two branches: one is a segment of a line, and the other is a two-dimensional surface. The wavefronts, at the caustic region, have self intersections and singularities. The ronchigrams exhibit closed-loop fringes when the grating is placed at the caustic region.

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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]
  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. A. A. Sherwood, “Quantitative analysis of the Ronchi test in terms of Ray optics,” J. Br. Astron. Assc. 68, 180–191 (1958).
  6. A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. N. S. W. 93, 19–23 (1959).
  7. D. Malacara and A. Cornejo, “Null Ronchi test for aspherical surfaces,” Appl. Opt. 13, 1778–1780 (1974).
    [CrossRef]
  8. T. Yatagai, “Fringe scanning Ronchi test for aspherical surfaces,” Appl. Opt. 23, 3676–3679 (1984).
    [CrossRef]
  9. A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Null Hartmann and Ronchi–Hartmann tests,” Appl. Opt. 29, 4618–4621 (1990).
    [CrossRef]
  10. 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]
  11. A. Cordero-Dávila, J. Daz-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]
  12. 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]
  13. M. V. Berry, “Disruption of images: the caustic-touching theorem,” J. Opt. Soc. Am. A 4, 561–569 (1987).
    [CrossRef]
  14. 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]
  15. 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]
  16. M. Mansuripur, “The Ronchi test,” Opt. Photon. News 8, 42–46 (1997).
  17. D. Malacara, “Geometrical Ronchi test of aspherical mirrors,” Appl. Opt. 4, 1371–1374 (1965).
    [CrossRef]
  18. V. I. Arnold, Catastrophe Theory (Springer-Verlag, 1986).
  19. V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhauser, 1995).
  20. V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, 1980).
  21. 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]
  22. D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
    [CrossRef]
  23. P. S. Theocaris, “Surface topography by caustics,” Appl. Opt. 15, 1629–1638 (1976).
    [CrossRef]
  24. P. S. Theocaris, Properties of caustics from conic reflectors. 1: meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
    [CrossRef]
  25. 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]
  26. 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]
  27. 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]
  28. 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]
  29. D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
    [CrossRef]

2010 (1)

2009 (1)

2008 (1)

2004 (1)

2002 (2)

A. Cordero-Dávila, J. Daz-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 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)

1997 (1)

M. Mansuripur, “The Ronchi test,” Opt. Photon. News 8, 42–46 (1997).

1992 (1)

1990 (1)

1987 (1)

1984 (1)

1981 (2)

1977 (1)

1976 (2)

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]

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)

1959 (1)

A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. N. S. W. 93, 19–23 (1959).

1958 (1)

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

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 Maps (Birkhauser, 1995).

Berry, M. V.

Burkhard, D. G.

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 4, 358–365 (2002).
[CrossRef]

Castro-Ramos, J.

Caulfied, H. J.

Cordero-Dávila, A.

Cornejo, A.

Cornejo-Rodriguez, A.

Daz-Anzures, J.

de Ita Prieto, O.

Friday, W.

Gusein-Zade, S. M.

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

Malacara, D.

Mansuripur, M.

M. Mansuripur, “The Ronchi test,” Opt. Photon. News 8, 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 4, 358–365 (2002).
[CrossRef]

Murty, M. V. R. K.

Román-Hernández, E.

Ronchi, V.

Santiago-Santiago, J. G.

Shealy, D. L.

Sherwood, A. A.

A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. N. S. W. 93, 19–23 (1959).

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.

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]

Theocaris, P. S.

Varchenko, A. N.

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

Velázquez-Castro, J.

Yatagai, T.

Appl. Opt. (17)

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

D. Malacara, A. Cornejo, and M. V. R. K. Murty, “Bibliography of various optical testing methods,” Appl. Opt. 14, 1065–1065 (1975).
[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, Properties of caustics from conic reflectors. 1: meridional rays,” Appl. Opt. 16, 1705–1716 (1977).
[CrossRef]

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

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]

A. Cordero-Dávila, A. Cornejo-Rodriguez, and O. Cardona-Nuñez, “Null Hartmann and Ronchi–Hartmann tests,” Appl. Opt. 29, 4618–4621 (1990).
[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. Daz-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, 6080–6088 (2004).
[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. 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 (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]

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

J. Proc. R. Soc. N. S. W. (1)

A. A. Sherwood, “Ronchi test charts for parabolic mirrors,” J. Proc. R. Soc. N. S. W. 93, 19–23 (1959).

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. Photon. News (1)

M. Mansuripur, “The Ronchi test,” Opt. Photon. News 8, 42–46 (1997).

Other (4)

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).

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

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Figures (13)

Fig. 1.
Fig. 1.

Schematic drawing of the Ronchi test arrangement. In this diagram we have the surface under test, locally given by z=f(x,y), a point light source located on the optical axis, and a Ronchi ruling. The pattern observed through the grating on the surface of the mirror is referred to as the real ronchigram.

Fig. 2.
Fig. 2.

Schematic drawing of the two sets of coordinate systems used to compute the image of a one-dimensional object obtained by reflection on an arbitrary smooth surface locally given by z=f(x,y). We also have included an emitted light ray such that its associated reflected light ray connects a point of the smooth surface with a point of the one-dimensional object. The point, on the smooth surface, where the emitted light ray is reflected belongs to the shadow of the one-dimensional object.

Fig. 3.
Fig. 3.

Caustic given by Eqs. (28) and (29) when the point light source is placed at (a) (0, 0, and 13.5 cm), (b) (2, 0, and 13.5 cm), and (c) (4, 0, and 13.5 cm).

Fig. 4.
Fig. 4.

Intersections of the caustic surface, given by Eqs. (28) and (29), with the plane y=0 when the point light source is placed at (a) (0, 0, and 13.5 cm), (b) (2, 0, and 13.5 cm), and (c) (4, 0, and 13.5 cm).

Fig. 5.
Fig. 5.

Intersections of some reflected wavefronts and the caustic surface, with the plane y=0 when the point light source is placed at (0, 0, and 13.5 cm).

Fig. 6.
Fig. 6.

Some reflected wavefronts when the point light source is placed at (4, 0, and 13.5 cm). Observe that the wavefronts are smooth surfaces out of the caustic region [(a) and (f)], while at the caustic region they are singular [(b)–(e)].

Fig. 7.
Fig. 7.

(a) Object space: the grating or Ronchi ruling and the caustic, which is a circle of radius Ra=0.0213cm and its center. (b) Image space: the associated Ronchigram. In this case s=(0,0,and13.5cm) and the grating is placed at the plane z=6.51cm.

Fig. 8.
Fig. 8.

(a)–(d) Object space and (e)–(h) image space for nd=(1.2)Ra,Ra,Ra/2, and 0, respectively. In this case s=(0,0,and13.5cm) and the grating is placed at the plane z=6.51cm.

Fig. 9.
Fig. 9.

(a) Object space: the grating or Ronchi ruling and the caustic, which is a circle of radius Rb=0.1142cm. (b) Image space: the associated Ronchigram. In this case s=(0,0,and13.5cm) and the grating is placed at the plane z=6.79cm.

Fig. 10.
Fig. 10.

(a)–(d) Object space and (e)–(h) image space for nd=(1.2)Rb,Rb,Rb/2, and 0, respectively. In this case s=(0,0,and13.5cm) and the grating is placed at the plane z=6.79cm.

Fig. 11.
Fig. 11.

(a) Intersections of the caustic surface, given by Eqs. (28) and (29), when the point light source is placed at s=(0,0,and13.5cm) and the planes z=constant (where the Ronchi ruling is placed) with the plane y=0. In (b)–(f) we present the ronchigrams when the grating is placed at the planes (b) z=6.23cm, (c) z=6.51cm, (d) z=6.79cm, (e) z=7.07cm, and (f) z=7.35cm, respectively.

Fig. 12.
Fig. 12.

(a) Intersections of the caustic surface, given by Eqs. (28) and (29), when the point light source is placed at s=(2,0,and13.5cm) and the planes z=constant (where the Ronchi ruling is placed) with the plane y=0. In (b)–(f) we present the ronchigrams when the grating is placed at the planes (b) z=6.23cm, (c) z=6.51cm, (d) z=6.79cm, (e) z=7.07cm, and (f) z=7.35cm, respectively.

Fig. 13.
Fig. 13.

(a) Intersections of the caustic surface, given by Eqs. (28) and (29), when the point light source is placed at s=(2,0,and13.5cm) and the planes z=constant (where the Ronchi ruling is placed) with the plane y=0. In (b)–(f) we present the ronchigrams when the grating is placed at the planes (b) z=6.23cm, (c) z=6.51cm, (d) z=6.79cm, (e) z=7.07cm, and (f) z=7.35cm, respectively. In this case the grating makes an angle π/4 with the Tx axis.

Equations (32)

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X(s,x,y)=r(x,y)+[C|r(x,y)s|]R^(s,x,y),
R^=hα
h1=(xs1)(1fx2+fy2)2fx[fy(ys2)+s3f],
h2=(ys2)(1+fx2fy2)2fy[fx(xs1)+s3f],
h3=(fs3)(1+fx2+fy2)+2[fx(xs1)+fy(ys2)],
α=(1+fx2+fy2)(s1x)2+(s2y)2+(s3f)2.
C=C±(x,y)|rs|+α(H1±H124H2H02H2),
H0=h·[(rx)×(ry)],H1=h·[(rx)×(hy)+(hx)×(ry)],H2=h·[(hx)×(hy)],
X=Xc±=r+(H1±H124H2H02H2)h.
X(x,y,z0)=x+[z0f(x,y)](h1(x,y,s)h3(x,y,s)),Y(x,y,z0)=y+[z0f(x,y)](h2(x,y,s)h3(x,y,s)),Z(x,y,z0)=z0,
X(x,y)=x+[z0f(x,y)](h1(x,y,s)h3(x,y,s)),Y(x,y)=y+[z0f(x,y)](h2(x,y,s)h3(x,y,s)).
Tx=Γ(σ),Ty=Σ(σ),
Ty=Λ(Tx).
Tx(σ)=x+[z0f(x,y)](h1(x,y,s)h3(x,y,s)),Ty(σ)=y+[z0f(x,y)](h2(x,y,s)h3(x,y,s))
Ty=TxtanΘ+nd,
y+[z0f(x,y)](h2(x,y,s)h3(x,y,s))={x+[z0f(x,y)](h1(x,y,s)h3(x,y,s))}tanΘ+nd.
f=a2x2y2,r=(x,y,a2x2y2),
h=(1f2)[2(s·r)ra2(s+r)],
α=|sr|a2f2,
R^=[2(s·r)ra2(s+r)]|sr|a2.
X(s,x,y)=r+[C|rs|][2(s·r)ra2(s+r)]|rs|a2.
X(0,x,y)=(2aC)r^.
H0=(a2f3)[(sr)·r],
H1=(2a2f5)(sr)·[a2(sr)+(s·r)r],
H2=(a4f7){|sr|2[3(s·r)a2]+s2[s·ra2]},
C=2|rs|[(sr)·r](2sr)·r,
C+=2|rs|32s23(s·r)+a2.
Xc=a2s(2sr)·r,
Xc+=2[a2s2(s·r)2]r+a2[s·ra2]sa2[a2+2s23(s·r)].
y+(z0f)(2(s·r)ya2(s2+y)2(s·r)fa2(s3+f))=[x+(z0f)(2(s·r)xa2(s1+x)2(s·r)fa2(s3+f))]tanΘ+nd.
y(2z0+a)=x(2z0+a)tanΘnda.
y=xtanΘnDwithDda2z0+a,

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