Quantitative evaluation of an off-axis parabolic mirrorby using a tilted null screen

Maximino Avendaño-Alejo, Victor I. Moreno-Oliva, Manuel Campos-García,*and Rufino Díaz-Uribe

Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México,Apdo. Postal 70-186, México City 04510, D.F. México

*Corresponding author: manuel.campos@ccadet.unam.mx

Received 11 November 2008; accepted 31 December 2008;posted 15 January 2009 (Doc. ID 103953); published 9 February 2009

We report the testing of a fast off-axis surface based on the null screen principles. Here we design a tiltednull screen with drop shaped spots drawn on it in such a way that its image, which is formed by reflectionon the test surface, becomes an exact square array of circular spots if the surface is perfect. Any departurefrom this geometry is indicative of defects on the surface. Here the whole surface is tested at once. Thetest surface has a radius of curvature of r ¼ 20:4mm (F=0:206). The surface departures from the bestsurface fit are shown; in addition, we show that the errors in the surface shape are below 0:4 μmwhen theerrors in the determination of the coordinates of the centroids of the reflected images are less than 1 pixel,and the errors in the coordinates of the spots of the null screen are less than 0:5mm. © 2009 OpticalSociety of America

OCIS codes: 220:4840, 220:1250, 120:6650.

1. Introduction

Among the principal methods for testing opticalsurfaces, the Ronchi and Hartmann tests have beenpopular for many years for testing slow (F=# > 1)spherical and aspherical surfaces [1,2]. However, itis possible to test a surface using Ronchi andHartmann methods with the same mathematicaltheory [3]; both are considered as geometric methods.The use of null lenses or computer generatedholograms (CGHs) to transform the spherical wave-front coming from a point source, to an asphericalwavefront that matches the test surface, is expensivewith respect to the Ronchi and Hartmann tests.According to Meinel and Meinel [4], the testing ofan off-axis parabolic mirror is best accomplished ifno additional optics is needed. Here we continue ear-lier work for testing off-axis concave surfaces [5,6] byusing a null screen test. This test (see Fig. 1) consistsof a screen with a uniform array of points alongperpendicular lines that through reflection we can

see at the CCD sensor a perfect square array ofpoints; in other words, each intersection point ofthe lines on the screen becomes a point source; afterreflection on the surface under test, the pencil of raysthat pass through a small aperture at the center ofthe screen (without taking into account diffractioneffects) forms the image of the screen, as in a cameraobscura. The aperture is large enough to avoid impor-tant diffraction effects, so this test is in the geometricoptics regime. A positive lens is used to collect andfocus these thin pencils of rays onto the CCD sensor.The effects of the aberrations introduced by the po-sitive lens of the camera have been well explained in[7] in which coma and spherical aberrations are con-sidered; the distortion is taken into account by cali-brating the camera lens; departures from a regulararray of points on the CCD or image are indicativeof deformations of the surface under test. In parti-cular, in [5] there is a proposal to test an off-axisparabolic surface by designing a plane null screenorthogonal to the local normal to the center ofthe surface, where it is simpler to align the off-axisparabolic mirror by reflection with respect to the cen-ter of the off-axis surface than the usual methods in

0003-6935/09/051008-08$15.00/0© 2009 Optical Society of America

1008 APPLIED OPTICS / Vol. 48, No. 5 / 10 February 2009

which the source is near to the focus of the parentsurface. In [6] an extension of the method is proposedfor testing an off-axis parabolic mirror by using tiltednull screens, near the tangential and sagittal caus-tics regions.The tilt allows us to control the size of the screen

and the sensitivity of the test; a preliminary qualita-tive test was shown. It is important to state that weexplored different tilted null screens near the tan-gential caustic surface for better alignment of the ex-perimental setup with drop shaped spots in bothradial and rectangular arrays. Here we evaluatequantitatively a very fast off-axis parabolic mirror(F=0:206) by using tilted null screens near the tan-gential caustic surface. This method is based onthe following procedure: we know a priori the idealor design parameters of the mirror under test; thisinformation is sufficient to design the tilted nullscreen as described in [6]. By using a CCD sensor,an image of the screen obtained by reflection onthe test surface is recorded; for each spot on theCCD sensor we calculate its centroid positions to findthe normals to the surface, and by an integration pro-cedure the shape of the surface under test is quanti-tatively obtained. We will explain the integrationmethod used and the errors introduced by it, andfinally we discuss some possible improvements ofthe test.

2. Theory

The procedure to obtain a null screen to test an off-axis surface was explained in earlier papers in greatdetail [5,6], but in order to make this paper self-contained, we describe briefly the steps to design atilted null screen. To calculate the ðx3; y3; z3Þ coordi-nates of the points on screen P3 that yield a perfectsquare grid on the CCD, we proceed backward, start-ing at the CCD plane (see Fig. 2). Given a point

ðx1; y1; aþ bÞ, aþ b is the distance between centralpoint V of the off-axis segment and the CCD plane.Here, x1 and y1 are the Cartesian coordinates of apoint on the CCD sensor; we trace a ray back throughhole P with coordinates ð0; 0; bÞ, where b is the dis-tance from point V to the hole aperture, as is shownin Fig. 2. We obtain the ðx; y; zÞ coordinates for eachpoint of incidence on the ideal surface by intersectingthe incident ray with the same surface. According to[8], the ideal shape of an off axis surface z can beexpressed as

zðx; yÞ ¼ γβ þ ðβ2 − αγÞ1=2 ; ð1Þ

with

α ¼ cð1þ k cos2θÞ;β ¼ ð1þ k × sin2θÞ−1=2 − ckx × sin θ × × cos θ;γ ¼ cð1þ k × sin2θÞx2 þ cy2; ð2Þ

where c is the curvature at the vertex of the parenton-axis surface, k is the conic constant, and θ is theangle between the optical or the symmetry axis of theparent surface and the z axis; furthermore, this anglecan be calculated by

tan θ ¼ cxc½1 − ðkþ 1Þc2x2c �1=2

; ð3Þ

where xc is the distance from the optical axis of theparent surface to the center of the segment. Thecoordinates ðx; y; zÞ for the point of incidence onthe off-axis surface are obtained by solving the simul-taneous equations

x − x1x1

¼ y − y1y1

¼ z − a − ba

: ð4Þ

The reflected ray is calculated through the vectorreflection law

R ¼ I − 2ðI · NÞN ¼ ðRx;Ry;RzÞ; ð5Þ

Fig. 1. Layout of the testing configuration.

Fig. 2. Normal evaluation (rays R and I do not necessarily lie inthe x–z plane).

10 February 2009 / Vol. 48, No. 5 / APPLIED OPTICS 1009

where I is the unit incident vector and N ¼ ðnx;ny;nzÞis given by

N ¼ ð−zx;−zy; 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2x þ z2y þ 1

q ¼ ð−∂z=∂x;−∂z=∂y; 1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið∂z=∂xÞ2 þ ð∂z=∂yÞ2 þ 1

p ; ð6Þ

the normal vector to the test surface.Finally we intersect the reflected ray with a tilted

plane outside the caustic region, where the nullscreen lies, as was explained in [6], in order to avoidsingularities. The plane in which we place the screenis given by

Apx3 þ Bpy3 þ Cpz3 ¼ Dp; ð7Þ

where ðAp;Bp;CpÞ are the director cosines for the nor-mal to the screen plane and Dp is a constant, so thatDp=Cp is the intersection of the plane with the z axis.For the surface described before through Eqs. (1)–(3),the screen plane is inclined in such a way thatBp ¼ 0; then the screen plane equation is reduced to

Apx3 þ Cpz3 ¼ Dp: ð8Þ

Imposing that Dp ¼ a, and Ap, Cp are the directioncosines in the X–Z plane and defining ϕ as the screentilt angle given by the relationship,

ϕ ¼ arctan�Cp

Ap

�: ð9Þ

Thus the intersection coordinates ðx3; y3; z3Þ on theplane are given by

x3 ¼ xþ RxMa;

y3 ¼ yþ RyMa;

z3 ¼ zþ RzMa; ð10Þ

where

Ma ¼�a − ðxþ z tanϕÞRx þ Rz tanϕ

�; ð11Þ

and Rx, Ry, and Rz are the Cartesian components ofvector R [Eq. (5)].In previous papers we designed null screens with

equal size dots, which produce unequal sized andshaped spot images at the CCD, making it difficultto process the image to evaluate the spot centroidsas has been done for convex aspherical surfaces[7]. In order to solve this problem, in [9] it was pro-posed to design each dot in such a way that all thespots at the CCD have a circular shape of equal size;the dot shape on the screen becomes like asymmetri-cal ovals; we call them drop shaped spots. Followingthis idea, we designed a tilted null screen in order totest an off-axis parabolic mirror with a square gridarray with drop shaped spots on a tilted null screen.

We define the spot image as the points ðxi; yiÞ onthe image around the ðxo; yoÞ points, such as fulfillingthe next condition: ðxo − xiÞ2 þ ðyo − yiÞ2 ≤ r2, where ris the circle’s radius with arbitrary length, centeredon the ðxo; yoÞ points as is shown in Fig. 1. We can alsoapply the ray tracing formulas given in [6] for eachsmall circle that has its center in ðxo; yoÞ. For conve-nience we chose the radius length to be a few pixels tofacilitate the processing of the image and to obtainthe centroid of each image to recover the shape ofthe surface under test.

3. Surface Shape Evaluation

The shape of the test surface can be obtained frommeasurements of the positions of the incident pointson the CCD plane through the formula [5]

z − zo ¼ −

Zp

po

�nx

nzdxþ ny

nzdy

�; ð12Þ

where zo is the sagitta for one point of the surfacethat must be known in advance. This expressionis exact.

A. Normal Evaluation

To evaluate the normals N to the test surface, we per-form a three-dimensional ray tracing procedure simi-lar to that used for evaluation of the normals for aconcave surface [10]. The procedure consists of find-ing the directions of the rays that join the actualpositions P0

1 ¼ ðx01; y01;aþ bÞ of the centroids ofthe spots on the CCD and their correspondingCartesian coordinates of the objects of the null screenP3 ¼ ðx3; y3; z3Þ.

According to the reflection law, the normal N to thesurface can be evaluated as

N ¼ R − IjR − Ij ; ð13Þ

where I and R are the directions of the incident andthe reflected rays on the surface, respectively (seeFig. 2). With reference to Fig. 2, the direction ofthe reflected ray R is known because after the reflec-tion on the surface it passes through the center of thelens stop at P and arrives at the CCD image plane atP0

1; this direction is given by

R ¼ ðx01; y01;aÞðx021 þ y021 þ a2Þ1=2 : ð14Þ

On the other hand, for the incident ray I we onlyknow point P3 at the null screen, so we have to ap-proximate a second point to obtain the direction ofthe incident ray by intersecting the reflected ray witha reference surface; in practice we choose the designsurface of the null screen as a reference surface, thusthe errors in the determination of the normals areminimal [7]. Then the reflected ray, whose direction

1010 APPLIED OPTICS / Vol. 48, No. 5 / 10 February 2009

is given by Eq. (14), intersects the reference surfaceat Ps ¼ ðxs; ys; zsÞ.Now, a straight line joining P3 with Ps gives

approximately the direction of the incident ray

I ¼ ðxs − x3; ys − y3; zs − z3Þ½ðxs − x3Þ2 þ ðys − y3Þ2 þ ðzs − z3Þ2�1=2

: ð15Þ

Finally, substituting Eqs. (14) and (15) intoEq. (13), the approximated normals to the test sur-face are calculated by

N ¼ ðδx; δy; δzÞðδ2x þ δ2y þ δ2z Þ1=2

; ð16Þ

with

δx ¼x01

ðx021 þ y021 þ a2Þ1=2

þ x3 − xs½ðxs − x3Þ2 þ ðys − y3Þ2 þ ðzs − z3Þ2�1=2

;

δy ¼y01

ðx021 þ y021 þ a2Þ1=2

þ y3 − ys½ðxs − x3Þ2 þ ðys − y3Þ2 þ ðzs − z3Þ2�1=2

;

δz ¼a

ðx021 þ y021 þ a2Þ1=2

þ z3 − zs½ðxs − x3Þ2 þ ðys − y3Þ2 þ ðzs − z3Þ2�1=2

: ð17Þ

4. Integration Procedure

Once the normals to the test surface are calculatedfrom the measurements of the centroids of theimages of the null screen, the next step is the numer-ical evaluation of Eq. (12). The common method forthe numerical evaluation is the trapezoidal rulefor nonequally spaced data [11],

zm ¼ −Xm−1

i¼1

��nxi

nzi

þ nxiþ1

nziþ1

� ðxiþ1 − xiÞ2

þ�nyi

nzi

þ nyiþ1

nziþ1

� ðyiþ1 − yiÞ2

�þ zo; ð18Þ

where m is the number of points along some integra-tion path.To analyze the details of the resulting surface eval-

uation, the data of the sagitta z were fitted to an off-axis conic surface given by [12]

z ¼ cxðx − xoÞ22

þ cyðy − yoÞ22

þ a3ðx − xoÞ2ðy − yoÞ þ a4ðy − yoÞ3þ a5ðx − xoÞ4 þ a6ðx − xoÞ2ðy − yoÞ2þ a7ðy − yoÞ4 þ Axþ Byþ zo; ð19Þ

where cx and cy are the curvatures at the origin ofcoordinates in the off-axis conic, and the coefficientsa3, a4, a5, a6, and a7 depend on the parameters de-fining the conic surface [12]. Here, ðxo; yoÞ is a decen-tering term, zo is the defocus, and A and B are theterms of tilt in x and y, respectively.

5. Error Analysis

Here we analyze the error obtained in the evaluationof the sagitta with the proposed method. In order tosimulate more realistic errors, we randomly displacethe coordinates of the centroids of the bright spots onthe CCD and the coordinates of the spots on the nullscreen [7]. The numerical simulation was performedconsidering the fast off-axis parabolic mirror, withthe design parameters given in Table 1.

For this task, we perform a numerical simulation,which consists of a three-dimensional ray traceprocedure (see Fig. 2). Thus starting at point P0

1on the CCD plane, we trace a ray, which passesthrough the small aperture lens stop at P and inter-sects the simulated surface at P2. Then, the actualnormals Na ¼ ðnxa;nya;nzaÞ to the surface are evalu-ated, according to

Na ¼ ∇f ðx; y; zÞj∇f ðx; y; zÞj jP2

; ð20Þ

where f ðx; y; zÞ represents the off-axis segment[Eq. (1)].

Next, we simulate random errors involved in thelocation of the centroid of each spot of the imageby applying a random function to the coordinatesP0

1 ¼ ðx01 þ δx; y01 þ δyÞ of the positions of thecentroids:

ðx01; y01Þ → ðx01 þ δx; y01 þ δyÞ: ð21Þ

The random displacement functions δx and δy weredeveloped to simulate Gaussian noise [13], and theyare given by

δx ¼ η2ð−2 ln r1Þ1=2 cosð2πr2Þ;

δy ¼ η2ð−2 ln r1Þ1=2 sinð2πr2Þ;

δz ¼ η2ð−2 ln r3Þ1=2 cosð2πr2Þ; ð22Þ

Table 1. Design Parameters for Testing the Off-Axis Mirror

Element Symbol Size

Surface radii of curvature r 20:4mmConic constant k −1:0Surface major diameter MD 49:4mmCamera lens focal length a 8mmCCD length d 6:6mmStop aperture—surface vertex distance b 210mmOffset xc 25:4mmTilt of the screen ϕ 30°

10 February 2009 / Vol. 48, No. 5 / APPLIED OPTICS 1011

where r1, r2, and r3 are uniformly distributed randomvariables that return values between 0 and 1, and η isa parameter that allows control of the size of the dis-placement functions.Then, once the displaced coordinates of the cen-

troids are obtained, the next step is to evaluatethe approximated normals N using Eq. (16). Thuswith the actual Na [Eq. (20)] and the approximatednormals N, the differences in sagitta can be obtainedin an approximate way from [7]

δz ≈ −

Z ��nxa

nza−nx

nz

�dxþ

�nya

nza−ny

nz

�dy

�; ð23Þ

where we have assumed that dxa ≈ dx (and dya ≈ dy).In Fig. 3, we show the plots of the rms differences

in sagitta δz for different values of parameter ηagainst the radial coordinate of the surface. Here,the squares represent the differences in sagitta forη ¼ 1 pixel and the circles η ¼ 0:5 pixel. The simula-tion was performed considering images with 20 spotsand the statistics over 72 simulated images. Fromthe plot we can observe that, as it must be expected,the differences in sagitta are smaller for displace-ment parameter η ¼ 0:5 pixel than for η ¼ 1:0 pixel.For η ¼ 1 pixel the differences in sagitta are smallerthan 0:32 μm; this means that if we can measure thecentroid of each spot on the CCD with an accuracybetter than 1 pixel, then the contribution to the un-certainty to the surface shape is lower than 0:32 μmfor all the evaluated points. If we are able, however,to find the centroids of the image spots with anaccuracy better than 0.5 pixel, then the contributeduncertainty on the surface evaluation is less than160nm or about λ=4 for λ ¼ 632:8nm.On the other hand, in order to find the error

contribution due to the errors on positioning of thespots of the null screen, we apply the coordinatesP3 ¼ ðx3; y3; z3Þ to the displacement functions in

Eq. (22). After that, we calculate the approximatenormals using Eq. (16) and the actual normals fromEq. (20); finally using Eq. (23) we obtain the rms dif-ferences in sagitta δz. The corresponding plot isshown in Fig. 4, here the squares correspond to va-lues of the displacement parameter η ¼ 1mm, andthe circles to η ¼ 0:5mm. Thus with the aim of mea-suring the sagitta to an accuracy better than 0:4 μm,we must locate the spots on the null screen with anerror less than 0:5mm, which is well inside the reso-lution for a 600 dpi laser printer.

6. Testing a Fast Concave Elliptical Mirror

A. Experimental Setup

We tested a fast off-axis parabolic mirror, which has adesign radius of curvature at the vertex of r ¼20:4mm (F=0:206); the rest of the design parametersare listed in Table 1. The images were captured witha CCD black-and-white camera (Sony Model XC-ST70), with a CCD sensor of 8:8mm × 6:6mm, andan 8mm focal length, with a Computar video lens at-tached. A flat screen was made on a laser printer onbond paper. This null screen was placed on an acrylicplane to give it mechanical strength. Although ambi-ent illumination may be enough to see the image, tohave better contrast, we illuminate the screen fromoutside with fluorescent lamps.

The test surface was mounted on two linear stagesfor easy centering along the x and y directions. Thenull screen was supported on a rotatory stage toincline the screen at the right angle. The CCD cam-era wasmounted on an x–y–z stage to locate it in sucha position that the entire surface can be observed;thus the whole surface can be evaluated at once(see Fig. 5). The alignment of the surface wasperformed manually by using a reference circle onthe image of the surface. The circular image of theboundary of the mirror must be centered at the

Fig. 3. Rms differences in sagitta obtained in a simulation whena random displacement is added to the coordinates of the centroidsof the spots at the CCD.

Fig. 4. Rms differences in sagitta obtained in a simulation whena random displacement is added to the coordinates of the positionsof the spots of the null screen.

1012 APPLIED OPTICS / Vol. 48, No. 5 / 10 February 2009

CCD andmust touch the upper and lower boundariesof the image. In addition, the image of the null screenmust show a square grid or array of points. If thiscondition is not fulfilled, then the screen is misa-ligned or the testing surface differs from the designsurface. The remaining misalignment can be com-puted by fitting the experimental data to Eq. (19).

B. Surface Evaluation

For a quantitative evaluation of the shape of the testsurface, we used a tilted null screen to produce asquare array of circular spots on the image plane.The spots are calculated with the aim of obtaininga set of well-defined white spots uniformly distribu-ted on a dark background and having all the samesize (0:02mm radius at the CCD) on the image plane[9]. Figure 6 shows how the screen looks before it isput in the experimental setup (screen flat). Note howthe spots on the screen have an almost ellipticalshape, but they are oriented along different direc-tions depending on its position. The image of the flatscreen after reflection on the off-axis mirror is shownin Fig. 7.

Making a first qualitative analysis of Fig. 7, we canobserve that, in general, the image shows analmost regular square grid pattern in the center ofthe image, except for the left zone of themirror wherethe square grid pattern is slightly curved, and it iseasy to determine that in this zone the mirrordeparts from the parabolic shape more significantly.Furthermore, some points, near the edge of the mir-ror, look blurred. The observed image deformationsare due to the tested surface and not to deformations

Fig. 5. Picture of the actual experimental setup.

Fig. 6. Tilted null screen.

Fig. 7. Image of the tilted null screen after reflection on the off-axis mirror.

Fig. 8. Calculated centroids of the image.

10 February 2009 / Vol. 48, No. 5 / APPLIED OPTICS 1013

of the paper screen or even misalignments of the testsurface. Then, the null screen method can be used todetect tiny shape deformations of the test surface.The centroids of the image are shown in Fig. 8;

they were calculated with the image-processing pro-gram ImageJ [14]. All the centroids were correctedfor the lens distortion (coefficient E ¼ −5:4204 ×10−9 mm−2). The next step is calculating the approxi-mated normals [Eq. (13)] to the test surface, asdescribed in Section 3. With the calculated normalsthe shape of the surface is obtained using Eq. (12)with the trapezoidal rule as the integration proce-dure [Eq. (18)]. The initial starting integration pointand some integration paths for the square grid areshown in Fig. 9. Some integration paths are very

large and introduce large numerical error in theintegration [Eq. (23)]. In Fig. 10 we show a recon-struction of the shape of the test surface.

To analyze the details of the evaluation we fit thedata to Eq. (19) by using the Levenberg–Marquardtmethod [15] for nonlinear least squares fitting that issuitable for this task. The results of the fits areshown in Table 2. It is well known that for all off-axisconics the curvatures are related by [16]

cyc2 ¼ c3x : ð24Þ

Here cx and cy are the sagittal and tangential curva-tures, respectively, and c is the curvature of the par-ent surface. From Table 2 the curvature of the parentsurface is 0:0525mm−1 and differs by approximately0:0035mm−1 or about 7% of the design value of0:0490mm−1 (see Table 1). In Fig. 11, we show thedifferences in sagitta between the evaluated surfaceand the best fitting off-axis conic. In this case the P −

V differences in sagitta between the evaluated pointsand the best fit is δzpv ¼ 0:020mm, and the rms dif-ference in sagitta value is δzrms ¼ 2:6 μm.

Then the null screen method allows measurementof the shape of the surface. Here, departures from theperfect shape have been clearly observed.

7. Conclusions

We have shown experimentally and in a quantitativeway that the proposal of using tilted flat null screensis useful for testing off-axis parabolic surfaces.

The introduction of random displacements in thecoordinates of the centroids on the CCD and in thecoordinates of the sources (null screen objects) al-lowed us to perform an error analysis of the method.We found that in order to have differences in sagittalower than 0:4 μm, the error in the measurement ofthe centroid coordinates must be less than 1 pixel,and the error in the measurement of the position

Fig. 9. Selected integration paths for the integration.

Fig. 10. Reconstruction of the test surface.

Table 2. Parameters Resulting from the Least Square Fitting of the Sagitta Data

cx ðmm−1Þ cy ðmm−1Þ xo ðmmÞ yo ðmmÞ zo ðmmÞ A

0.064 0.095 −0:742 0.047 −0:166 −2:914B a3 ðmm−2Þ a4 ðmm−2Þ a5 ðmm−3Þ a6 ðmm−3Þ a7 ðmm−3Þ0.024 −4:3 × 10−6 2:0 × 10−5 −1:0 × 10−6 −5:3 × 10−6 1:8 × 10−6

Fig. 11. Differences in sagitta between the measured surface andthe best fitting off-axis conic.

1014 APPLIED OPTICS / Vol. 48, No. 5 / 10 February 2009

of the sources of the null screen must be lessthan 0:5mm.For the quantitative evaluation we found that the

measurement performed with the trapezoidal meth-od gives a curvature that differs approximately 7%from the design curvature. The result of the leastsquare fit shows that the tested surface is very closeto the design surface except for the borders where thedifferences are larger.It is important to take into account that with this

testing method the shape of the parabolic off-axissurface can be evaluated with accuracy of approxi-mately 2 μm rms. An improved algorithm for obtain-ing subpixel resolution in the evaluation of thecentroids of the image spots will also help to improvethe accuracy of the method. This test, as it was im-plemented in this paper, is very useful to test veryfast off-axis surfaces where the traditional tests can-not be easily implemented.

The authors of this paper are indebted to NeilBruce for his help in revising the manuscript. Thisresearch was supported by the Consejo Nacionalde Ciencia y Tecnología under project U51114-Fand Proyectos de Investigacion e Innovacion Techno-logica under project ES-114507.

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