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Measurement and correction of misalignments in corneal topography using the null-screen method

Manuel Campos-García, Daniel Aguirre-Aguirre, Victor Ivan Moreno-Oliva, Oliver Huerta-Carranza, and Victor de emanuel armengol-cruz

Open Access Open Access

Abstract

Misalignments are a common problem in corneal topography reconstruction. To determine the topography of the surface using a corneal topographer based on the null-screen method, it is necessary to measure errors caused by mechanical misalignment of the topographer, which could influence the results of this technique but are not related to the quality of the corneal surface. To evaluate the variability in measurements, in this work, we simulate some misalignments in the optical system, for the simulation we design a semi-radial null-screen to test a reference spherical surface to identify the variations in results introduced by misalignment errors of the corneal topographer with respect to the test surface. According to the simulations, the accuracy of the null-screen method ranges from 0.81 µm to 2.84 µm for typical tilt, decentering, and defocusing errors. Experimental results for the testing of a spherical surface are shown. After removing the misalignments, we found that the variations are approximately 3.5 μm rms value measured with respect to the best-fitting sphere, and the radius of curvature differs approximately 0.06% from the design value.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The null-screen method has been used to test fast and symmetric surfaces with great success [1–3], but in recent years this method has been applied to obtain the corneal topography [4,5]. The cornea has an essential role in human vision. It has approximately two thirds of the eye’s total refractive power. In its normal state, the anterior surface of the cornea has an average radius of curvature of approximately 7.8 mm in the center. An irregular shape of the cornea may produce some refractive errors such as myopia, hyperopia, or astigmatism. It is important, for some clinical and research applications, to obtain a detailed assessment of the cornea with great accuracy.

The current methods for evaluating the corneal surface are mainly based on Placido’s disk, a plate with concentric bright and dark rings. These rings are reflected over the surface of the cornea and the virtual image is acquired by a camera. Then, by image processing and numerical evaluation, the algorithms find the intersections of the rings and the meridian. The local radius can be found by calculating the distances of the intersections of neighboring rings, the corneal height can be calculated by a numerical integration along the meridians and, by taking the first and second derivative, the curvature can be obtained. The topographers based on Placido’s disk have a tolerance of ±0.02 mm in the measurement of the curvature radius and the rms error of the heights is in the range of 2-6 μm [6–8].

As pointed out, in previous works, we proposed measuring the corneal topography using a conical null-screen and have made the first evaluation of a calibration sphere surface [4] and some human corneal surfaces [5]. However, to evaluate the variability in measurements due to systematic errors, in this paper we simulate image patterns considering some misalignments in the optical system, and we obtain the topography and the geometric parameters of the simulated surface. The results obtained provide us with information on the sensitivity and accuracy of the test.

Additionally, we implement an experimental design of the null-screen technique to measure the topography of a reference surface, it is necessary to determine the effects of tilt, decentering and defocusing errors caused by mechanical misalignment, which could influence the results of this technique but are not related to the shape of the surface under test.

2. Null-screen method

The null-screen is an array of custom targets [1–5], which after being reflected by a surface will produce a specific pattern on the image plane. For fast convex surfaces, as the anterior corneal surface, the testing configuration is show in Fig. 1. The procedure for the design of the null-screen is based on the ray-tracing method described in [5], and we shall summarize it in this section for the sake of clarity of exposition. Here, each target P3 is reflected by the anterior corneal surface at P2. After reflection on the corneal surface, the pencil of rays that pass through a small aperture at P, forms the image of the screen on the image plane at P1, as in a camera obscura.

 figure: Fig. 1.

Fig. 1. Variables involved in the conical null-screen design.

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To determine the target points P3 on the conical null-screen that give us a custom array of circular points on the image plane after the reflection from the test surface it is necessary to select each element from the pattern and perform an inverse exact ray-tracing calculation. We start with the position of the elements in the image plane P1 = (ρ1, ϕ1, -a-b-e), where a is the distance from the pinhole to the CCD/CMOS plane, b is the distance from the pinhole to the cone base, and e is the distance from the vertex of the surface to the cone base (a, b, e > 0), see Fig. 1. Assuming that the anterior corneal surface is a conic with symmetry of revolution, which is described by

$$f(x,y,z) = Q{z^2} - 2rz + {\rho ^2}, $$
where z is the sagitta, ρ = (x2 + y2)1/2 is the distance of each point of the surface to the optical axis z, r is the radius of curvature at the vertex, and Q = k + 1 (k is the conic constant of the surface). Then, the incident ray I that passes through the aperture at P = (0, 0, -b-e) reaches the anterior corneal surface at the point P2 = (ρ2, ϕ1+π, z2), with
$$\begin{array}{l} {\rho _2} = \frac{{a\{{Q({b + e} )+ r} \}- {{[{{{({ar} )}^2} - ({b + e} )\rho_{_1}^2\{{Q({b + e} )+ 2r} \}} ]}^{{1 / 2}}}}}{{Q{a^2} + \rho _{_1}^2}}{\rho _1},\\ {z_2} = \frac{{a\{{Q({b + e} )+ r} \}- {{[{{{({ar} )}^2} - ({b + e} )\rho_{_1}^2\{{Q({b + e} )+ 2r} \}} ]}^{{1 / 2}}}}}{{Q{a^2} + \rho _{_1}^2}}a - ({b + e} ). \end{array}$$

Finally, the coordinates of the targets in the null screen are obtained by the intersection of the reflected ray R with the cone at the point P3 = (ρ3, ϕ1+π, z3), with

$$\begin{array}{l} {\rho _3} = \frac{{s\left\{ {\left( {\frac{{{\rho_1}\rho_2^2 - {\rho_3}{{({Q{z_2} - r} )}^2} + 2a{\rho_2}({Q{z_2} - r} )}}{{a\rho_2^2 - a{{({Q{z_2} - r} )}^2} - 2{\rho_1}{\rho_2}({Q{z_2} - r} )}}} \right)({{z_2} + e + h} )+ {\rho_2}} \right\}}}{{\left( {\frac{{{\rho_1}\rho_2^2 - {\rho_3}{{({Q{z_2} - r} )}^2} + 2a{\rho_2}({Q{z_2} - r} )}}{{a\rho_2^2 - a{{({Q{z_2} - r} )}^2} - 2{\rho_1}{\rho_2}({Q{z_2} - r} )}}} \right)h + s}},\\ {z_3} = \frac{{h\left\{ {\left( {\frac{{{\rho_1}\rho_2^2 - {\rho_3}{{({Q{z_2} - r} )}^2} + 2a{\rho_2}({Q{z_2} - r} )}}{{a\rho_2^2 - a{{({Q{z_2} - r} )}^2} - 2{\rho_1}{\rho_2}({Q{z_2} - r} )}}} \right){z_2} + {\rho_2}} \right\} - s({e + h} )}}{{\left( {\frac{{{\rho_1}\rho_2^2 - {\rho_3}{{({Q{z_2} - r} )}^2} + 2a{\rho_2}({Q{z_2} - r} )}}{{a\rho_2^2 - a{{({Q{z_2} - r} )}^2} - 2{\rho_1}{\rho_2}({Q{z_2} - r} )}}} \right)h + s}}. \end{array}$$

From Fig. 1, we can easily see that the diameter of the reflected image d on the CCD/CMOS sensor is

$$d = \frac{{aD}}{{b + e + \beta }},$$
where D is the diameter of the test surface, and β is the sagitta at the rim of the surface, which for a conical surface Eq. (1) is given by
$$\beta = \frac{r}{Q}\left[ {1 - {{\left( {1 - \frac{{Q{D^2}}}{{4{r^2}}}} \right)}^{{1 / 2}}}} \right]. $$

3. Surface shape evaluation method

The shape of the test surface can be obtained from measurements of the positions of the incident points P1 on the CCD/CMOS plane through the formula

$$z - {z_i} = \int\limits_{{P_i}}^{{P_f}} {\sqrt {{{\left( {\frac{{{n_x}}}{{{n_z}}}} \right)}^2} + {{\left( {\frac{{{n_y}}}{{{n_z}}}} \right)}^2}} d\rho } ,$$
where zi is the sagitta for one point of the surface that must be known in advance (the value of zi is not obtained from the test; for discrete point evaluation, this is approximated by Eq. (3) at point (x1, y1) as is explained below), and N = (nx, ny, nz) is the normal to the corneal front surface. This expression is exact, but must be solved numerically, evaluating the normal and performing the numerical integration, which is an approximation, and introduces some errors that must be reduced. It is important to mention, that the form of Eq. (6) is better when the integral is evaluated numerically, since the errors involved in its evaluation are substantially reduced because we have one integral and not two, as in the previous integral expression [2–5]. The most common method used for the discrete evaluation of the integral of Eq. (6) is the trapezoidal rule for nonequally spaced data
$$z - {z_i} \approx \sum\limits_{k = 1}^N {\left[ {\left\{ {\sqrt {{{\left( {\frac{{{n_{{x_{k + 1}}}}}}{{{n_{{z_{k + 1}}}}}}} \right)}^2} + {{\left( {\frac{{{n_{{y_{k + 1}}}}}}{{{n_{{z_{k + 1}}}}}}} \right)}^2}} + \sqrt {{{\left( {\frac{{{n_{{x_k}}}}}{{{n_{{z_k}}}}}} \right)}^2} + {{\left( {\frac{{{n_{{y_k}}}}}{{{n_{{z_k}}}}}} \right)}^2}} } \right\}\frac{{\left( {\sqrt {x_{k + 1}^2 + y_{k + 1}^2} - \sqrt {x_k^2 + y_k^2} } \right)}}{2}} \right]}, $$
where N is the number of points along a custom integration path, and zi is approximated by Eq. (3) at point (x1, y1).

The evaluation of the normal vectors to the test surface can be performed with an approximate iterative algorithm as is described in [5]. The proposed algorithm involves three-dimensional ray tracing. The procedure consists of finding the directions of the rays that join the real positions P1 = (x1, y1, -a-b-e) of the centroids of the spots on the CCD/CMOS and the corresponding Cartesian coordinates of the targets of the null-screen P3 = (x3, y3, z3), see Fig. 2. To evaluate the approximated normals N’ we must calculate the directions of the incident I’ and reflected rays R’. The direction of the reflected ray R’ is known because after the reflection on the surface it passes through the center of the lens stop at P and arrives at the image plane at P1; this ray is given by

$${{\bf{R}}^{\prime}} = \frac{{({{x_1},{y_1}, - a} )}}{{{{({x_1^2 + y_1^2 + {a^2}} )}^{{1 / 2}}}}}. $$

This reflected ray R’ intersect the anterior corneal surface at Ps = (xs, ys, zs), which is described by a general aspheric surface

$$z = \frac{{r - {{\{{{r^2} - Q[{{{({x - {x_{\rm{o}}}} )}^2} + {{({y - {y_{\rm{o}}}} )}^2}} ]} \}}^{{1 / 2}}}}}{Q} + {\kern 1pt} {T_x}({x - {x_{\rm{o}}}} )+ {T_y}({y - {y_{\rm{o}}}} )+ {z_{\rm{o}}}, $$
where (xo, yo, zo) are the coordinates of the vertex of the surface; (xo, yo) are the decentering terms, zo is the defocusing, and Tx and Ty are the terms of tilt in x and y directions, respectively. Then, from the target points P3 and those on the corneal surface Ps, the incident ray I’ is
$${{\bf{I}}^{\prime}} = \frac{{({{x_s} - {x_3},\;{y_s} - {y_3},\;{z_s} - {z_3}} )}}{{{{[{{{({{x_s} - {x_3}} )}^2} + {{({{y_s} - {y_3}} )}^2} + {{({{z_s} - {z_3}} )}^2}} ]}^{{1 / 2}}}}}. $$

Finally, according to the reflection law, the approximated normal vectors N’ can be evaluated as

$${{\bf{N}}^{\prime}} = \frac{{{{\bf{R}}^{\prime}} - {{\bf{I}}^{\prime}}}}{{|{{{\bf{R}}^{\prime}} - {{\bf{I}}^{\prime}}} |}}$$

 figure: Fig. 2.

Fig. 2. Schematic diagram that shows the evaluation of the normal vectors.

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After, with the approximated normals we obtain the shape of the surface through Eq. (7). Next, the data of the sagitta is fitted to Eq. (9) to obtain a new reference surface, and with this we calculate the new approximated normal and so on until arrive to the shape of the surface [5].

It is important to mention that in practice the initial reference surface is calculated with a randomized algorithm [9]. This is done by proposing a general conic surface; the surface parameters are randomly generated, and the calculated null-screen is compared against the originally designed screen. The coefficients that generate the null-screen closest to the reference null-screen are taken as the final values that describe the surface under test. Our randomized algorithm has three basic steps: two cycles to find the nearest solution to the problem, and a third cycle, only for the acquisition of the entrance data.

4. Systematic error analysis

To obtain measurements with high accuracy by using the null-screen method, an accurate alignment between null-screen, CCD/CMOS camera, and test surface is critical. For this we perform a systematic error analysis to find both the minimal error that can be detected, and the accuracy of the method. The analysis focuses on the misalignments of both the null-screen, assuming that the test surface and the CCD/CMOS camera are fixed and aligned to each other relative to the coordinate system, and misalignments of the test surface assuming that the null-screen and the CCD/CMOS camera are fixed and aligned. The different states of misalignment of the null-screen and surface are divided into translations and rotations [10] and for simplicity they are analyzed separately. These misalignments of the null-screen and the surface are treated independently because that allow to find the minimal error that can be detected with the proposed method. Additionally, for the simulation we design a semi-radial null-screen to test a spherical reference surface with radius of curvature 7.8 mm and diameter of 12 mm. In Table 1, the null-screen design parameters are shown.

Tables Icon

Table 1. Conical null-screen design parameters.

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4.1 Null-screen misalignments

To determinate the effect of the null-screen misalignments in the evaluation of the corneal topography, we perform a procedure like the development in Ref. [11] to analyze misalignment errors in the evaluation of parabolic trough solar collector (PTSC). So, we shall summarize the procedure here for the sake of clarity of exposition. According to [11], the translations and rotations for the null-screen points P3 are given, respectively, by

$${\bf P}{{\bf ^{\prime}}_3} = {{\bf P}_3} + \delta {\bf T}, $$
and
$${\bf P}{{\bf ^{\prime}}_3} = R \times ({{{\bf P}_3} - {\bf V}} )+ {\bf V}, $$
where δT = (δx, δy, δz) is a translation vector, V = (0, 0, -h) are the coordinates of the null-screen vertex, and R represents the rotation matrix around the each one of the Cartesian axis [11].

On the one hand, assuming that the coordinates of a misaligned null-screen are given by P’3 = (x’3, y’3, z’3). We start at P’3 and calculate the direction of the incident I’ and reflected ray R’ to find the positions P’1 = (x’1, y’1, -a-b-e) of the new grid points on the image plane. Here the components of the incident ray are simply given by

$${\bf I^{\prime}} = \frac{{{\bf P}{{\bf ^{\prime}}_2} - {\bf P}{{\bf ^{\prime}}_3}}}{{|{{\bf P}{{\bf^{\prime}}_2} - {\bf P}{{\bf^{\prime}}_3}} |}} = ({I{^{\prime}_x},I{^{\prime}_y},I{^{\prime}_z}} ), $$
and the components of the reflected ray are given by
$${\bf R^{\prime}} = \frac{{{\bf P} - {\bf P}{{\bf ^{\prime}}_2}}}{{|{{\bf P} - {\bf P}{{\bf^{\prime}}_2}} |}} = ({R{^{\prime}_x},R{^{\prime}_y},R{^{\prime}_z}} ). $$

However, the incident I’ and reflected ray R’ depend on the unknown point P’2 = (x’2, y’2, z’2) that can be calculated numerically from

$$\frac{{N{^{\prime}_x}}}{{N{^{\prime}_z}}} = \frac{{I{^{\prime}_x} - {R^{\prime}}_x}}{{I{^{\prime}_z} - {R^{\prime}}_z}},\quad \frac{{N{^{\prime}_y}}}{{N{^{\prime}_z}}} = \frac{{I{^{\prime}_y} - {R^{\prime}}_y}}{{I{^{\prime}_z} - {R^{\prime}}_z}},$$
together with Eq. (1) [11].

Finally, we calculate the departures between ideal P1 and deformed grid points P’1 on the detection plane (see Fig. 3), these departures are given by

$$\Delta {\bf{P}} = \sqrt {{{({{x_1} - {x_1}^{\prime}} )}^2} + {{({{y_1} - {y_1}^{\prime}} )}^2}} . $$

The departures calculated with Eq. (17) give the sensitivity of the test because if we assume that the minimum detectable deviation on the detection plane is equal to the size of 1 pixel, we can readily find the minimum detectable misalignment on the null-screen with the setup proposed in this work. To analyze the variations introduced in the measurements, the simulation was performed with different values of misalignments of the null-screen.

 figure: Fig. 3.

Fig. 3. Departures from the ideal positions for a perfect null-screen.

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The Figs. 4(a-c) show the plots of ΔPrms values against the misalignments along the x, y and z-axis, respectively. Here, the decentering and defocusing values ranging from 0 to 230 μm in steps of 23 μm. The plots show a quadratic behavior, increasing according to the misalignment of the null-screen along the corresponding axis. From the plots of Fig. 4.a-b) we can see that if the minimum detectable variation in the misalignment along the x-axis and y-axis, respectively, from the perfect centroid position on the CMOS image plane is equal to the size of 1 pixel, in our case 1.25 µm, we can find that the minimum misalignment error that can be detected is smaller than 117.93 µm; and the minimum defocusing error along the z-axis that can be measure is 131.03 µm, see Fig. 4.c). These values result from the intersection of the horizontal line with the data curves. This is a measure of the sensitivity of the test to these misalignments, values below these misalignment errors cannot be detected with the proposed test.

 figure: Fig. 4.

Fig. 4. Rms deviations in the detection plane from a perfect grid as function of misalignments of the null-screen along the: a) x-axis, b) y-axis, c) z-axis. Rms differences in sagitta against the variations of the positions of the centroid for misalignments along the: d) x-axis, e) y-axis, d) z-axis.

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To evaluate the accuracy of the proposed test due to the different states of misalignment of the null-screen, we performed the calculation of the normals for each misalignment position of the null-screen and evaluated the shape of the surface from Eq. (7). The set of points are fitted to Eq. (9) to obtain the best-fitting surface. The fit of Eq. (9) was performed by using the Levenberg–Marquart method [12] for nonlinear least-squares fitting that is suitable for this task. In Figs. 4(d-f) we show the plots of the rms differences in sagitta between the evaluated surface and the best-fitting surface Δzrms against the rms departures ΔPrms from the ideal grid positions. The relation of the rms variations has a quadratic behavior. Again, assuming that the minimum detectable deviation from the perfect centroid position on the image is equal to the size of 1 pixel on the CCD/CMOS sensor, we can find that the minimum deviation error that can be detected is smaller than 1.05 μm for misalignments along the x and y-axis; and smaller than 2.84 μm for the z-axis. This an interesting result, because shows that the defocusing misalignment introduces more error in the evaluation of the surface.

The Figs. 5(a-c) shows the plots of the rms deviations on the detection plane ΔPrms as a function of the rotation angle, which behaves quadratically. From the plots it is easy to see that the minimum rotation angle about the x and y-axis that can be measured with the proposed test is 2 mrad, and 1.45 mrad for rotations about the z-axis. Rotations of the null-screen below these values cannot be detected. Next, in Figs. 5(d-f) plots of the rms differences in sagitta Δzrms against the deviation on the detection sensor plane are shown. These plots give us information on the accuracy of the test; in particular, if we detect variations equal to 1 pixel, the minimum value in the sagitta error that can be attained with this is test is 1.22 μm for rotations about the x, y-axis, and 0.98 μm for the z-axis.

 figure: Fig. 5.

Fig. 5. Rms deviations on the detection plane from a perfect grid for rotation error of the null-screen about the: a) x-axis, b) y-axis, c) z-axis. Rms differences in sagitta against the variations of the positions of the centroid for rotation about the: d) x-axis, e) y-axis, f) z-axis.

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4.2 Surface misalignments

The misalignment of the surface can be performed by changing the decentering terms (xo, yo), the defocusing zo, and the tilt terms Tx and Ty in x and y in Eq. (9), respectively. Then, with the aim of calculating the departure points P’1 = (x’1, y’1, -a-b-e) with respect to the ideal grid points P1 = (x1, y1, -a-b-e) obtained from the reflected image of the misaligned surface, Eq. (17), see Fig. 6, we calculated the points P’2 on the misaligned surface by solving numerically Eqs. (16) and (9). From Fig. 6, we have that the incident ray to the misaligned surface is given by

$${\bf I^{\prime}} = \frac{{{\bf P}{{\bf ^{\prime}}_2} - {{\bf P}_3}}}{{|{{\bf P}{{\bf^{\prime}}_2} - {{\bf P}_3}} |}} = ({I{^{\prime}_x},I{^{\prime}_y},I{^{\prime}_z}} ), $$
and the reflected ray is given by Eq. (15). Both rays depend on the unknown point P’2 of the assumed misaligned surface.

 figure: Fig. 6.

Fig. 6. Departures from the ideal positions for a perfect and well-positioned surface.

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Then, the reflected ray R’ hits the CCD/CMOS detection plane at points P’1 giving us the reflected image of the misaligned surface, see Fig. 6. Finally, we calculated the departures between ideal and deformed grid points through Eq. (17). Then, assuming that the minimum detectable deviation from the perfect point on the observation plane is equal to the size of 1 pixel, we can calculate the minimum deviation on the test surface that can be detected with the proposed method. To analyze the variations introduced in the measurements, the simulation was performed with different values of misalignments of the test surface.

For the misalignments of the test surface, the plots of Fig. 7.a-c) show the rms departures ΔPrms from the ideal positions for a perfect and well-positioned surface for different values of transverse and axial decentering along z-axis. The plots show a linear behavior, increasing according to the misalignment of the test surface along the corresponding axis. From the plots of Fig. 7.a-b) we can see that if the minimum detectable variation in the misalignment along the x-axis and y-axis from the perfect centroid position on the image plane is equal to the size of 1 pixel, the minimum misalignment error that can be detected is smaller than 0.28 µm; and the minimum error along the z-axis that can be measured with the proposed test is 0.24 µm, see plot of Fig. 7.c). These values result from the intersection of the horizontal line with the data curves. As before, this is a measure of the sensitivity of the test to such misalignments. Values below these misalignments cannot be detected with the proposed test.

 figure: Fig. 7.

Fig. 7. Rms deviations in the detection plane from a perfect grid as function of misalignments of the surface along the: a) x-axis, b) y-axis, c) z-axis. Rms differences in sagitta against the variations of the positions of the centroid for misalignments along the: d) x-axis, e) y-axis, d) z-axis.

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As before, to evaluate the accuracy of the proposed test due to the misalignments of the surface, we performed the evaluation of the surface for each misalignment position. In Figs. 7(d-f) we show the plots of the rms differences in sagitta Δzrms against the rms departures ΔPrms from the ideal positions. The linear relation gives us information on the variations of the positions of the centroids with the rms differences in sagitta Δzrms. Here, we can readily find that the minimum rms differences in sagitta that can be detected is smaller than 0.81 μm for misalignments along the x-axis, 0.82 μm along y-axis, and smaller than 1.02 μm for the z-axis, being the defocusing the largest error that must be corrected.

Finally, in the plots of Figs. 8(a-b) we show the results when we perform tilts on the x-axis and y-axis, respectively, on the test surface. Here, we can see a linear relation between the rms deviation on the detection plane against the tilt values. From the plots it is easy to see that the minimum tilt value about the x and y-axis that can be measured with the proposed test is 5.41 × 10−3. Tilt values of the surface below these values cannot be detected. And in Figs. 8(c-d) the plots of the rms differences in sagitta against the deviation on the detection sensor plane are shown. In both plots, we see that the minimum value in the rms differences in sagitta that can be attained with this is test is 0.83 μm and 0.82 μm for tilt about the x, and y-axis, respectively.

 figure: Fig. 8.

Fig. 8. Rms deviations on the detection plane from a perfect grid for tilt of the surface about the: a) x-axis, b) y-axis. Differences in sagitta against the variations of the positions of the centroid for tilt about the: c) x-axis, d) y-axis.

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It is important to note that the sensitivity and accuracy depend on the CCD/CMOS sensor resolution. The above results are for an error of 1 pixel in the sensor plane, for subpixel accuracy the sensitivity and the accuracy of the method could be increased, our algorithm for the centroid recovery has an accuracy of a tenth of a pixel. This is important in optical component manufacturing since we could know in advance what the technical specifications of the components would be in the experimental setup to be below the quality specifications required by the optical design.

The simulated results suggest that the misalignment errors of the null-screen technique are significant and must be quantified. For an experimental case, the evaluated sagitta values will be fit to a general conical surface (Eq. (9)) where the misalignment errors appear explicit. After quantifying the misalignment errors, they must be considered in order to compute the true value position of the null-screen (Eq. (13)) and the sagitta values (Eq. (7)).

5. Experimental results

We performed a quantitative test of a spherical reference surface diameter of 12 mm and radius of curvature of 7.8 mm. To capture the reflected images, we used a CMOS camera sensor (EO-18112, Edmund Optics) with a sensitive area of 6.1 × 4.6 mm (4912 × 3684 pixels), pixel size of 1.25 µm, and a lens 25-mm focal length attached. In this case, the lens diaphragm was used as an aperture stop. The CMOS camera was in a position such that the entire surface could be observed, so that almost the entire surface could be evaluated at once. The null-screen was designed to produce a semi-radial array of circular spots on the image plane, see Section 3. Each object spot was designed in such a way that it had a circular shape of equal size at the CMOS (0.01 mm radius); the circular dot on the detection plane becomes an asymmetrical oval on the null-screen when reflected off the test surface, due to the mapping of circles onto the curved surface of the sphere.

The alignment of the optical system was performed manually by using an overlay that consists of a reference circle and a cross hair target drawn on the image of the surface. The circular image of the boundary of the surface must be centred at the CMOS. Hence, the image of the null-screen must show a semi-radial array of circular spots. If this condition is not fulfilled, then, there is a misalignment of the optical system, or the testing surface is different to the design surface. The developed method used for the analysis of experimental patterns follows the process shown in the flowchart of Fig. 9.

 figure: Fig. 9.

Fig. 9. Flowchart of the surface analysis method.

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The image of the conical null-screen after reflection on the spherical surface is shown in Fig. 10.a). The centroids of the image in Fig. 10.a) were calculated with an image-processing program using custom algorithms. This algorithm obtains the region of interest of the original image, and then we calculated the local maximum of each spot. Then, we construct a rectangle around each spot and evaluate the coordinates of the centroids using statistical averaging [13]. Finally, all the centroids were corrected for the lens distortion [14].

 figure: Fig. 10.

Fig. 10. a) Reflected image of the null-screen by the reference sphere, b) elevation map for the reference sphere.

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According to Section 3, we evaluated the misaligned reference surface and the misaligned null-screen with the randomized algorithm [9] developed by the authors. In Table 2, the decentering coefficients of the surface under test (xo, yo, zo) and the null screen (xo3, yo3, zo3) recovered by our algorithm are shown. From the Table 2, we see that the rms differences in sagitta between the theoretical surface and the surface retrieved by our algorithm is 8.2 μm.

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Table 2. Decentering coefficients recovered by the randomized algorithm.

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The next step is to calculate the approximated normal [Eq. (11)] to the test surface, as described in Section 3. With the calculated normal, the topography of the surface is obtained using Eq. (7).

In Table 3 we show the geometrical parameters of the test surface resulting from the least square fit to the data obtained from the sagitta of the evaluated reference surface (Eq. (7)). In this case, the rms difference in sagitta between the evaluated points and the best fit value is Δzrms = 3.5 μm. Additionally, we notice that the radius of curvature differs by 5 µm or about 0.06% of the design value of r = 7.80 mm. In the plot of Fig. 10.b), we show the elevation map obtained from the differences in the sagitta between the measured surface and the best fitting sphere obtained by a least-squares fit. Here decentering, defocusing, and tilt are automatically removed because the experimental sagitta values not only have the misalignments of the optical system but also the deformations of the test surface, and the data from the best fitting sphere have the misalignments calculated (see Table 3); then when the differences in sagitta are performed the misalignments are cancelled.

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Table 3. Parameters resulting from least squares fitting of sagitta data.

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6. Conclusions

We presented a theoretical analysis to describe and quantify the sensitivity and accuracy of the corneal topographer based on conical null-screen through misalignments of the null-screen and the reference surface. The results showed that small values of tilt, decentering, and defocusing errors produce variation in the coordinates of the reflected rays.

Here we found that assuming that the minimum detectable deviation from the perfect centroid position on the image is equal to the size of 1 pixel on the detector plane, we can readily find that the minimum values for misalignment error that can be detected with the proposed method are: for misalignment of the null-screen along the x-axis and y-axis 117.93 µm, along the z-axis 131.03 µm, for rotations of the null-screen around the x-axis and the y-axis 2 mrad and for the z-axis 1.45 mrad. For the misalignment of the surface along the x-axis and the y-axis 0.28 µm, along the z-axis 0.24 µm, for tilt of the surface along the x-axis and the y-axis 5.41 × 10−3. Finally, according to the simulations, the accuracy of the proposed testing method ranges from 0.81 µm to 2.84 µm, these results are comparable with some other works reported in the literature (2 to 6 μm) [6–8], being the defocusing, the largest error obtained.

For the quantitative evaluation, we found that the measurement performed with the null-screen method gives a radius of curvature that differs approximately 0.06% from the design radius of curvature and an accuracy of 3.5 μm in the shape of the surface. This procedure is able to evaluate tilt, position, and defocusing errors to correct the experimental results.

Funding

Consejo Nacional de Ciencia y Tecnología (293411); Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (PE101120, TA100319, TA100519).

Acknowledgments

The authors of this paper are indebted to Neil Bruce (Instituto de Ciencias Aplicadas y Tecnología, Universidad Nacional Autónoma de México, México) for his help in revising the manuscript

Disclosures

The authors declare no conflicts of interest.

References

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2. M. Campos- García, R. Bolado-Gómez, and R. Díaz-Uribe, “Testing fast aspheric concave surfaces with a cylindrical null screen,” Appl. Opt. 47(6), 849–859 (2008). [CrossRef]  

3. 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(5), 1008–1015 (2009). [CrossRef]  

4. M. Campos-García, C. Cossio-Guerrero, V. I. Moreno-Oliva, and O. Huerta-Carranza, “Surface shape evaluation with a corneal topographer based on a conical null-screen with a novel radial point distribution,” Appl. Opt. 54(17), 5411–5419 (2015). [CrossRef]  

5. M. Campos-García, V. de E. Armengol-Cruz, and A. I. Osorio-Infante, “Evaluating the anterior corneal surface using an improved null-screen system,” OSA Continuum 2(3), 736–748 (2019). [CrossRef]  

6. M. Kaschke, K. H. Donnerhacke, and M. S. Rill, Optical Devices in Ophthalmology and Optometry. Technology, Design Principles, and Clinical Applications, (Wiley-VCH, 2014).

7. M. Collins, S. Vincent, and S. Read, “The cornea,” in Handbook of visual optics, Vol. 1, P. Artal, ed. (CRC press, 2017).

8. W. Tang, M. J. Collins, L. Carney, and B. Davis, “The Accuracy and Precision Performance of Four Videokeratoscopes in Measuring Test Surfaces,” Optom. Vis. Sci. 77(9), 483–491 (2000). [CrossRef]  

9. D. Aguirre-Aguirre, R. Diaz-Uribe, M. Campos-Garcia, B. Villalobos-Mendoza, R. Izazaga-Pérez, and O. Huerta-Carranza, “Fast conical surfaces evaluation with null-screens and randomized algorithms,” Appl. Opt. 56(5), 1370–1382 (2017). [CrossRef]  

10. J. Pfund, N. Lindlein, and J. Schwider, “Misalignment effects of the Shack–Hartmann sensor,” Appl. Opt. 37(1), 22–27 (1998). [CrossRef]  

11. M. Campos-García, A. Peña-Conzuelo, O. Huerta-Carranza, J. R. Díaz-Uribe, U. E. Espinoza-Nava, and V. I. Moreno-Oliva, “Testing the surface quality of a reflective parabolic trough solar collector with two flat null-screens,” Appl. Opt. 58(4), 752–763 (2019). [CrossRef]  

12. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd ed. (McGraw-Hill, 1992), pp. 161–166.

13. P. Arulmozhivarman, L. Praveen Kumar, and A. R. Ganesan, “Measurement of moments for centroid estimation in Shack–Hartmann wavefront sensor—a wavelet-based approach and comparison with other methods,” Optik 117(2), 82–87 (2006). [CrossRef]  

14. C. Ricolfe-Viala and A. J. Sánchez-Salmerón, “Lens distortion models evaluation,” Appl. Opt. 49(30), 5914–5928 (2010). [CrossRef]  

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Figures (10)
Fig. 1.
Fig. 1. Variables involved in the conical null-screen design.

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Fig. 2.
Fig. 2. Schematic diagram that shows the evaluation of the normal vectors.

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Fig. 3.
Fig. 3. Departures from the ideal positions for a perfect null-screen.

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Fig. 4.
Fig. 4. Rms deviations in the detection plane from a perfect grid as function of misalignments of the null-screen along the: a) x-axis, b) y-axis, c) z-axis. Rms differences in sagitta against the variations of the positions of the centroid for misalignments along the: d) x-axis, e) y-axis, d) z-axis.

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Fig. 5.
Fig. 5. Rms deviations on the detection plane from a perfect grid for rotation error of the null-screen about the: a) x-axis, b) y-axis, c) z-axis. Rms differences in sagitta against the variations of the positions of the centroid for rotation about the: d) x-axis, e) y-axis, f) z-axis.

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Fig. 6.
Fig. 6. Departures from the ideal positions for a perfect and well-positioned surface.

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Fig. 7.
Fig. 7. Rms deviations in the detection plane from a perfect grid as function of misalignments of the surface along the: a) x-axis, b) y-axis, c) z-axis. Rms differences in sagitta against the variations of the positions of the centroid for misalignments along the: d) x-axis, e) y-axis, d) z-axis.

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Fig. 8.
Fig. 8. Rms deviations on the detection plane from a perfect grid for tilt of the surface about the: a) x-axis, b) y-axis. Differences in sagitta against the variations of the positions of the centroid for tilt about the: c) x-axis, d) y-axis.

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Fig. 9.
Fig. 9. Flowchart of the surface analysis method.

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Fig. 10.
Fig. 10. a) Reflected image of the null-screen by the reference sphere, b) elevation map for the reference sphere.

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Tables (3)

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Table 1. Conical null-screen design parameters.

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Table 2. Decentering coefficients recovered by the randomized algorithm.

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Table 3. Parameters resulting from least squares fitting of sagitta data.

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Equations (18)

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f ( x , y , z ) = Q z 2 − 2 r z + ρ 2 ,
ρ 2 = a { Q ( b + e ) + r } − [ ( a r ) 2 − ( b + e ) ρ 1 2 { Q ( b + e ) + 2 r } ] 1 / 2 Q a 2 + ρ 1 2 ρ 1 , z 2 = a { Q ( b + e ) + r } − [ ( a r ) 2 − ( b + e ) ρ 1 2 { Q ( b + e ) + 2 r } ] 1 / 2 Q a 2 + ρ 1 2 a − ( b + e ) .
ρ 3 = s { ( ρ 1 ρ 2 2 − ρ 3 ( Q z 2 − r ) 2 + 2 a ρ 2 ( Q z 2 − r ) a ρ 2 2 − a ( Q z 2 − r ) 2 − 2 ρ 1 ρ 2 ( Q z 2 − r ) ) ( z 2 + e + h ) + ρ 2 } ( ρ 1 ρ 2 2 − ρ 3 ( Q z 2 − r ) 2 + 2 a ρ 2 ( Q z 2 − r ) a ρ 2 2 − a ( Q z 2 − r ) 2 − 2 ρ 1 ρ 2 ( Q z 2 − r ) ) h + s , z 3 = h { ( ρ 1 ρ 2 2 − ρ 3 ( Q z 2 − r ) 2 + 2 a ρ 2 ( Q z 2 − r ) a ρ 2 2 − a ( Q z 2 − r ) 2 − 2 ρ 1 ρ 2 ( Q z 2 − r ) ) z 2 + ρ 2 } − s ( e + h ) ( ρ 1 ρ 2 2 − ρ 3 ( Q z 2 − r ) 2 + 2 a ρ 2 ( Q z 2 − r ) a ρ 2 2 − a ( Q z 2 − r ) 2 − 2 ρ 1 ρ 2 ( Q z 2 − r ) ) h + s .
d = a D b + e + β ,
β = r Q [ 1 − ( 1 − Q D 2 4 r 2 ) 1 / 2 ] .
z − z i = ∫ P i P f ( n x n z ) 2 + ( n y n z ) 2 d ρ ,
z − z i ≈ ∑ k = 1 N [ { ( n x k + 1 n z k + 1 ) 2 + ( n y k + 1 n z k + 1 ) 2 + ( n x k n z k ) 2 + ( n y k n z k ) 2 } ( x k + 1 2 + y k + 1 2 − x k 2 + y k 2 ) 2 ] ,
R ′ = ( x 1 , y 1 , − a ) ( x 1 2 + y 1 2 + a 2 ) 1 / 2 .
z = r − { r 2 − Q [ ( x − x o ) 2 + ( y − y o ) 2 ] } 1 / 2 Q + T x ( x − x o ) + T y ( y − y o ) + z o ,
I ′ = ( x s − x 3 , y s − y 3 , z s − z 3 ) [ ( x s − x 3 ) 2 + ( y s − y 3 ) 2 + ( z s − z 3 ) 2 ] 1 / 2 .
N ′ = R ′ − I ′ | R ′ − I ′ |
P ′ 3 = P 3 + δ T ,
P ′ 3 = R × ( P 3 − V ) + V ,
I ′ = P ′ 2 − P ′ 3 | P ′ 2 − P ′ 3 | = ( I x ′ , I y ′ , I z ′ ) ,
R ′ = P − P ′ 2 | P − P ′ 2 | = ( R x ′ , R y ′ , R z ′ ) .
N x ′ N z ′ = I x ′ − R ′ x I z ′ − R ′ z , N y ′ N z ′ = I y ′ − R ′ y I z ′ − R ′ z ,
Δ P = ( x 1 − x 1 ′ ) 2 + ( y 1 − y 1 ′ ) 2 .
I ′ = P ′ 2 − P 3 | P ′ 2 − P 3 | = ( I x ′ , I y ′ , I z ′ ) ,
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