Profilometry of toroidal surfaces with an improved Ronchi test

Santiago Royo, Josep Arasa, and Carles Pizarro

Santiago Royo, Josep Arasa, and Carles Pizarro

^{}The authors are with the Center for Development of Sensors, Instrumentation and Systems (CD6), Technical University of Catalonia, Violinista Vellsolà 37, E-08222 Terrassa, Spain.

An implementation of the well-known Ronchi test technique, which
allows for the profilometric measurement of nonrotationally symmetrical
surfaces, is presented and applied to the measurement of toroidal
surfaces. Both the experimental setup and the data-processing
procedures are described, and parameters such as the radius of
curvature of the sample surface, the orientation of its principal
meridians, and the position of its vertex are measured by means of the
values of the local normal to the surface obtained at a set of sampling
points. Integration of these local normal values allows for the
reconstruction of the three-dimensional profile of the toroidal surface
considered with micrometric accuracy, and submicrometric surface
details may be calculated by use of surface-fitting procedures. The
density of sampling points on the surface may be tailored to fit test
requirements, within certain limits that depend on selection of
experimental setup.

J. Arasa, S. Royo, C. Pizarro, “Toroidal surface profilometries through Ronchi deflectometry: constancy under rotation of the sample,” in Applications of Photonic Technology, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 909–915 (1998).

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross-grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

W. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

S. Royo, “Topographic measurements of non-rotationally symmetrical concave surfaces using Ronchi deflectometry,” Ph.D. dissertation (Technical University of Catalonia, Terrassa, Spain, 1999).

1999 (1)

H. J. Lee, S. W. Kim, “Precision profile measurement of aspheric surfaces by improved Ronchi test,” Opt. Eng. 38, 1041–1047 (1999).
[CrossRef]

J. Arasa, S. Royo, C. Pizarro, “Toroidal surface profilometries through Ronchi deflectometry: constancy under rotation of the sample,” in Applications of Photonic Technology, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 909–915 (1998).

W. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

Mobsby, E.

E. Mobsby, “A Ronchi null test for paraboloids,” Sky Telesc. 48, 325–330 (1974).

J. Arasa, S. Royo, C. Pizarro, “Toroidal surface profilometries through Ronchi deflectometry: constancy under rotation of the sample,” in Applications of Photonic Technology, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 909–915 (1998).

Royo, S.

J. Arasa, S. Royo, C. Pizarro, “Toroidal surface profilometries through Ronchi deflectometry: constancy under rotation of the sample,” in Applications of Photonic Technology, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 909–915 (1998).

S. Royo, “Topographic measurements of non-rotationally symmetrical concave surfaces using Ronchi deflectometry,” Ph.D. dissertation (Technical University of Catalonia, Terrassa, Spain, 1999).

Stahl, H. P.

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross-grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

W. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

Stultz, K.

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross-grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

D. S. Wan, M. W. Chang, “Effects of grating spacing on the Ronchi test,” Opt. Eng. 32, 1084–1090 (1993).
[CrossRef]

H. J. Lee, S. W. Kim, “Precision profile measurement of aspheric surfaces by improved Ronchi test,” Opt. Eng. 38, 1041–1047 (1999).
[CrossRef]

Sky Telesc. (1)

E. Mobsby, “A Ronchi null test for paraboloids,” Sky Telesc. 48, 325–330 (1974).

Other (6)

M. Jalie, The Principles of Ophthalmic Lenses (Association of British Dispensing Opticians, London, 1980).

S. Royo, “Topographic measurements of non-rotationally symmetrical concave surfaces using Ronchi deflectometry,” Ph.D. dissertation (Technical University of Catalonia, Terrassa, Spain, 1999).

W. Meyers, H. P. Stahl, “Contouring of a free oil surface,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 84–94 (1992).

A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing, 2nd ed., D. Malacara, ed. (Wiley, New York, 1992).

J. Arasa, S. Royo, C. Pizarro, “Toroidal surface profilometries through Ronchi deflectometry: constancy under rotation of the sample,” in Applications of Photonic Technology, G. A. Lampropoulos, R. A. Lessard, eds., Proc. SPIE3491, 909–915 (1998).

K. Stultz, H. P. Stahl, “A discussion of techniques that separate orthogonal data produced by Ronchi cross-grating patterns,” in Current Developments in Optical Design and Optical Engineering, R. E. Fischer, W. J. Smith, eds., Proc. SPIE2263, 226–232 (1994).

Intersection of 1-pixel-wide line patterns: (a)
nonmicrostepped experiment yielding 68 sampling points on the sample
surface; (b) microstepped experiment yielding 7950 sampling points
on the sample surface.

G60 orientation: three-dimensional profile of the
surface obtained by integration of the local normals: (a)
three-dimensional plot, (b) contour plot. Each contour step
equals a height step of 16 µm.

Residuals obtained by subtraction of the measured profile
from the best-fit profile. Each contour step equals 47
nm. (a) G00 orientation, (b) G30 orientation, (c) G60
orientation, (d) G90 orientation. Features that deviate from
the ideal surface may be seen to rotate as the sample is rotated.

C_{
J
} stands for the
curvature at the meridian considered, K_{
J
} for the
independent term of the linear fit, and R_{
J
} for
the radius of curvature. J may be either the base or the
cross curves of the surface, and r^{2} is the
correlation coefficient of the linear fit.

Table 2

Orientation G60: Results of the Three-Dimensional
Fitting of a Tilted and Decentered Spherocylindrical Surface (Eq.
3) to the Measured Profile

The position of the vertex was set
manually, so no reference value is
provided. R_{
B
} stands for the radius of the
base curve, R_{
C
} for the radius of the cross
curve, θ for the orientation of the principal meridians,
x_{0} and y_{0} for the
coordinates of the vertex of the surface, and r^{2}
for the correlation coefficient of the surface fit.

Table 3

Results for the Sample in the Three Additional
Orientations Measured

N stands for the number of
sampling points, A for the area sampled,
R_{
B
}^{LINEAR} for the radius of the base
curve obtained with linear fitting,
R_{
C
}^{LINEAR} for the radius of the cross
curve obtained with linear fitting,
r_{
B
}^{2(LINEAR)} and
r_{
B
}^{2(LINEAR)} for the correlation
coefficients of the linear fits for the base and the cross curves,
R_{
B
}^{3D} for the radius of the base
curve obtained with surface fitting,
R_{
C
}^{3D} for the radius of the cross
curve obtained with surface fitting, θ for the orientation of the
principal meridians, x_{0} and
y_{0} for the coordinates of the vertex of the
surface, r^{2(3D)} for the correlation
coefficient of the surface fit.

Tables (3)

Table 1

G60 Orientation: Results of Fitting the Plots in Fig. 6 to Eq. (5)

C_{
J
} stands for the
curvature at the meridian considered, K_{
J
} for the
independent term of the linear fit, and R_{
J
} for
the radius of curvature. J may be either the base or the
cross curves of the surface, and r^{2} is the
correlation coefficient of the linear fit.

Table 2

Orientation G60: Results of the Three-Dimensional
Fitting of a Tilted and Decentered Spherocylindrical Surface (Eq.
3) to the Measured Profile

The position of the vertex was set
manually, so no reference value is
provided. R_{
B
} stands for the radius of the
base curve, R_{
C
} for the radius of the cross
curve, θ for the orientation of the principal meridians,
x_{0} and y_{0} for the
coordinates of the vertex of the surface, and r^{2}
for the correlation coefficient of the surface fit.

Table 3

Results for the Sample in the Three Additional
Orientations Measured

N stands for the number of
sampling points, A for the area sampled,
R_{
B
}^{LINEAR} for the radius of the base
curve obtained with linear fitting,
R_{
C
}^{LINEAR} for the radius of the cross
curve obtained with linear fitting,
r_{
B
}^{2(LINEAR)} and
r_{
B
}^{2(LINEAR)} for the correlation
coefficients of the linear fits for the base and the cross curves,
R_{
B
}^{3D} for the radius of the base
curve obtained with surface fitting,
R_{
C
}^{3D} for the radius of the cross
curve obtained with surface fitting, θ for the orientation of the
principal meridians, x_{0} and
y_{0} for the coordinates of the vertex of the
surface, r^{2(3D)} for the correlation
coefficient of the surface fit.