What is a Hartmann test? Daniel Malacara-Hernández* and Daniel Malacara-Doblado Centro de Investigaciones en Optica, A. C., Loma del Bosque 115, León, Gto., Mexico *Corresponding author: [email protected] Received 22 October 2014; revised 5 February 2015; accepted 6 February 2015; posted 10 February 2015 (Doc. ID 225473); published 13 March 2015
In this paper we will review some of the many different practical arrangements that have been obtained to measure the transversal aberrations of optical systems based on the odd and vulnerable Hartmann test. There are many optical testing configurations that apparently are not related to the original Hartmann test. However, they are really the same thing and can be considered just a variation of the same basic arrangement, as will be described here. © 2015 Optical Society of America OCIS codes: (260.0260) Physical optics; (120.0120) Instrumentation, measurement, and metrology; (120.3930) Metrological instrumentation. http://dx.doi.org/10.1364/AO.54.002296
1. Introduction
The Hartmann test was invented by Hartmann in Germany more than a century ago [1]. Surprisingly, it is still in wide use today. The reason is its great simplicity and the sensitivity it has for measuring optical systems of many kinds. What we would like to point out here is that there are many optical testing configurations that apparently are not related to the original Hartmann test, but they are really the same thing and can be considered just a variation of the same basic arrangement. The mathematical analysis is the same. They have in common that the experimental measurements are the slopes of the surface or wavefront under measurement. In many systems this is equivalent to measuring the transverse aberration. 2. Measurements of the Transverse Aberrations and Integration to Retrieve the Wavefront
Let us consider a wavefront with an almost spherical shape, which can be aspheric. For example a plane wavefront is just an spherical wavefront with an infinite radius of curvature. The relation between the wavefront slope’s transverse aberrations and the wavefront deformations is surprisingly simple and 1559-128X/15/092296-06$15.00/0 © 2015 Optical Society of America 2296
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accurate (see Fig. 1). The exact relation was described by Rayces [2]: ∂Wx; y and ∂x ∂Wx; y TAy −r − Wx; y ; ∂y
TAx −r − Wx; y
(1)
but the wavefront deformations are so small compared with the radius of curvature of the wavefront that these expressions can be written as TAx −r
∂Wx; y ∂x
and TAy −r
∂Wx; y : ∂y
(2)
The integration methods for these two expressions in order to retrieve the wavefronts are many and have been the subject of research for more than a hundred years. All of these methods can broadly be classified as zonal or modal methods. The modal methods do not integrate the positions of the spots, as determined by their centroids, one by one as do the zonal methods, like that by Ghozeil and Simmons [3] or by Southwell [4]; instead, they integrate the pattern with the spots as a whole, assuming that the wavefront deformation can be represented by a combination of linearly independent analytical functions. These functions can be called aberrations and can
The selection of the wavefront analysis, modal or zonal, is quite important and depends on the type of wavefront aberration that is to be measured. If we expect to find small extent irregularities and low order aberrations, a zonal procedure is convenient. On the other hand, if high order polynomial aberrations are to be detected, a modal processing is more convenient. In some cases it may be necessary to analyze the wavefront with both methods. The detailed procedure should also take into account the pupil shape or any possible central obscuration. 3. Different Configurations Fig. 1. Relation between the wavefront deformations and the transverse aberrations or wavefront slopes.
Fig. 2. Classical Hartmann arrangement proposed in 1900.
have many different representations. Frequently, Zernike polynomials are used in the modal methods to represent the wavefront deformations. Modal methods started with the seminal work by Cubalchini [5] and then were described by Gavrielides [6] and later by Aksenov and Isaev [7], by Acosta et al. [8], and by Prieto et al. [9]. Another example of modal methods is based on Fourier techniques, as shown by Freischlad and Koliopoulos [10] and Canovas and Ribak [11], which can avoid the need for the centroid determination.
The Hartmann test has been so successful that many different configurations had been devised to measure wavefront or surface slopes and from those to retrieve the wavefront deformations. The classical arrangement, proposed by Hartmann in 1900 [1], is shown in Fig. 2, where the mirror or lens under test is covered with a screen with holes, and the transverse aberrations are measured by the deviations of the spots generated by each hole on the Hartmann pattern. Thus, the sampling plane is at the exit pupil of the system or at the mirror, and the observing plane is near, but not exactly at, the focus, so the spots are defocused. For this reason the measured positions are the centroids of the spots. If the mirror is spherical and the light source is close to the center of curvature of the mirror, the spot array at the observation plane is identical to the holes’ square array at the mirror, only scaled down. Hence, if the Hartmann screen is a square array of holes, the spots in the Hartmann plate also form a square array. If the mirror is an ellipsoid and the light source is at one of the foci we also have a square array at the Hartmann plate if the Hartmann screen is a square array of holes. We say that we have a null test if the array of spots at the Hartmann plate is a perfect square array of spots, when there is no undesired aberration at the optical system. If the mirror is aspheric and the light source is at the paraxial center of curvature of the mirror, the spots are along curved lines. This is not a null test. However, we may have a null test if the holes at the Hartmann screen have a distribution such that the array of spots at the
Fig. 3. Shack–Hartmann test. (a) Arrangement proposed by Platt and Shack in 1971 and (b) testing of collimated wavefronts. 20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS
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Fig. 4. Modified Hartmann test, analogous to a Ronchi test.
Hartmann plate is perfectly square when the mirror is a perfect aspherical surface. An extremely interesting and useful modification was proposed by Platt and Shack [12] in 1971, in order to measure collimated (flat) wavefronts, as in Fig. 3. With this arrangement we can measure collimated wavefronts in lenses or telescope mirrors, or even the wavefront aberrations of the human eye, as described by many researchers, for example by Hoffer et al. [13]. The main differences with the basic Hartmann test are that a collimated wavefront with a small diameter can be measured and that the spots are focused and their diameters are determined by diffraction. If the Shack–Hartmann lens array is a square array, as in most cases, we have a null test only for flat or slightly spherical wavefronts.
The Ronchi test [14], as the Hartmann test, measures the transverse aberrations, but only in one direction, perpendicular to the lines in the Ronchi ruling. If the ruling is substituted by a dark screen with a rectangular array of holes, as in Fig. 4, the arrangement can be considered a modified Ronchi test, but also a modified Hartmann test. We may also think in the superposition of two lateral shear interferometers, with shears in two orthogonal directions. If the magnitude of the lateral shear is small, conceptually this test is identical to a Ronchi test with two crossed rulings or to another modification of the Hartmann test. An important difference with the classic Hartmann test is that the sampling and observing planes are interchanged. An observing eye or camera has to be used to observe the virtual images of the spots located at the exit pupil or at the mirror. An interesting application of the same principle has been reported by Clare and Lane [15]. Null test configurations are like in the classic Hartmann test. Another possible configuration is by placing a small aperture imaging lens at the point conjugate to the virtual object position, as in Fig. 5. In front of the lens, near the front focal plane, an array of light sources is located. This array of light sources can be a LED array or even a nonluminous object with a reticle drawn on it. This arrangement can be imagined as a modification of the previous one, based on the Ronchi test, where the sampling plane (the array of light sources) is at the front of the entrance pupil of the system and not at the back of the system, after the exit pupil. The observing eye (Ronchi ruling) or camera should focus on the lens
Fig. 5. Testing with an array of small light sources.
Fig. 6. Testing of a lens with a screen with a rectangular array of lines. (a) Aspheric lens and (b) lens with spherical aberration. 2298
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or on the exit pupil of the optical system under test. Then, as a consequence, the spots are not focused on the measuring plane, as in the original Hartmann test. The small imaging lens and the virtual object are at conjugate positions, where the lens is assumed to be working. As before, the array of light sources can be replaced by a screen with a rectangular array of lines, and the imaging lens by the naked eye as in the testing of a lens in Fig. 6, as described by López-Ramírez et al. [16]. This is a useful tool for aspheric lenses. A null
test configuration is obtained when the virtual object position and the imaging lens are at the proper aberration-free conjugate positions for which the lens was designed. Other possible modified arrangements for null tests of concave paraboloidal mirrors or for concave spherical mirrors with ruled nonluminous screens are illustrated in Fig. 7 and described by Malacara-Doblado and Ghozeil [17]. These previous configurations can have several variations; for example, the lens under test does not necessarily have to be a convergent optical system. A divergent lens, for example an ophthalmic lens, as shown by Salas-Peimbert et al. [18], can be examined as in Fig. 8(a). The imaging system or observing eye should focus on the lens (or exit pupil) and not the light sources, so the spots are not necessarily in focus. This is convenient in order to associate each spot to its corresponding lens position. A flat glass window can be examined as in Fig. 8(b). Car glass windshields windows can be evaluated with this arrangement. The observing eye or camera should be focused on the glasses window under test, producing slightly defocused spots on the observing plane. However, a large error is not introduced if the array of light sources is focused instead of the glass window. With this configuration all the images of the light sources on the ellipsoid are in a plane, very close to the strong convex surface being measured. Since the diameter of the convex surface is small compared with the length of the ovoid, the positions of the virtual image of the light sources can be considered as being at this convex surface. An additional advantage of this configuration is that the spots are focused
Fig. 8. (a) Testing a divergent lens and (b) testing a glass window with a rectangular array of light sources.
Fig. 9. (a) Testing of a convex reflecting surface or a convex mirror with an array of light sources and (b) testing a convex mirror with a cylindrical ruled screen.
Fig. 7. Arrangements to test (a) a paraboloid and (b) a sphere with a ruled screen.
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Fig. 10. Testing with an array of light sources, placing their images (spots) all at approximately the same position.
on the detector as the observing plane. A strong convex surface, for example the corneal surface of the eye, can be measured as in Fig. 9(a). If the virtual observation surface with the images of the light sources is desired to be a flat surface, the array of light sources has to be over an ovoidal surface, as shown by Mejia-Barbosa and Malacara-Hernández [19]. This configuration has been used to measure the corneal topography of the eye. Instead of the ellipsoidal surface with the light sources, a cylindrical surface with the light sources or a dark screen with an array of lines can be used, as shown in Fig. 9(b). The spots in the image are not exactly in a plane, and not all the sampling surface with the light sources or lines array is in perfect focus. But since the center of gravity of the spots does not move, the accuracy of the instrument is not seriously affected as shown by Díaz-Uribe and Campos-García [20]. The array of light sources or of lines cannot form a square array if a null test is desired. The proper array configuration to be used should be calculated by ray tracing, in order to obtain a square array and hence a null test. A similar arrangement with the line arrays on a flat surface or flat computer display has been used with success with the name SCOTS to test telescope mirrors by Su et al. [21,22]. Another arrangement, where the spots are all at the same position, can be devised. The advantage is that the detector can be much smaller, increasing the sensitivity of the measurements. However, since the spots are superimposed at a single position, the light sources are turned on one by one and not all at the same time, as in Fig. 10. This system has been used in some commercial lens meters [23,24]. The focusing lens has the purpose that the cones of light arriving to the lens to be measured are all parallel to the optical axis in order to measure its back focal length. The position sensing detector is conjugate to the collimator and stop in front of the array of light sources. 4. Some Comments about the Transverse Aberration Measurements and the Reference Wavefront
The transverse aberrations are the deviations of the centroids of the spots with respect to their reference positions, with components TAx and TAy . After integration, the results represent the wavefront deformations with respect to the reference wavefront. There are several possibilities for the reference positions for the spots: (a) When the position for the reference spots is a single point at the optical axis. Then the retrieved 2300
APPLIED OPTICS / Vol. 54, No. 9 / 20 March 2015
wavefront represents the deformations with respect to a reference sphere with a different curvature than that of the wavefront, with center of curvature at the defocused observing plane. (b) If we have a null test, that is, if the unaberrated system produces a perfect square array of spots, and the position for the reference spots is obtained by generating a perfect array of spots identical to the sampling screen but scaled to a size that fits as much as possible to the measured array of spots, using a least squares procedure. Then the reference wavefront is a sphere with a curvature nearly equal to that of the wavefront. (c) If we have a nonnull test, the positions for the reference spots have to be calculated by ray tracing through the ideal optical system. Then, the retrieved wavefront represents the deformations with respect to the ideal wavefront, which could be spherical or aspherical. 5. Conclusions
In spite of being quite an old method, the Hartmann test is so powerful and simple that it is still being used for many types of systems. The great variety of possible different configurations is a consequence of its simplicity and usefulness. It is expected that these tests will be used with success for many decades more. References 1. J. Hartmann, “Bemerkungen uber den Bau und die Justirung von Spektrograpen,” Zt. Instrumentenkd. 20, 47–58 (1900). 2. J. L. Rayces, “Exact relation between wave aberration and ray aberration,” Opt. Acta 11, 85–88 (1964). 3. I. Ghozeil and J. E. Simmons, “Screen test for large mirrors,” Appl. Opt. 13, 1773–1777 (1974). 4. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980). 5. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,” J. Opt. Soc. Am. 69, 972–977 (1979). 6. A. Gavrielides, “Vector polynomials orthogonal to the gradient of Zernike polynomials,” Opt. Lett. 7, 526–528 (1982). 7. V. P. Aksenov and Y. N. Isaev, “Analytical representation of the phase and its mode components reconstructed according to the wave-front slopes,” Opt. Lett. 17, 1180–1182 (1992). 8. E. Acosta, S. Bará, M. A. Rama, and S. Ríos, “Determination of phase mode components in terms of local wave-front slopes: an analytical approach,” Opt. Lett. 20, 1083–1085 (1995). 9. P. M. Prieto, F. Vargas-Martín, S. Goelz, and P. Artal, “Analysis of the performance of the Hartmann–Shack sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000). 10. K. R. Freischlad and C. L. Koliopoulos, “Modal estimation of a wavefront from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986). 11. C. Canovas and E. N. Ribak, “Comparison of Hartmann analysis methods,” Appl. Opt. 46, 1830–1935 (2007).
12. B. C. Platt and R. V. Shack, “Lenticular Hartmann screen,” Opt. Sci. Newsl. 5, 15–16 (1971). 13. H. Hoffer, P. Artal, B. Singer, J. L. Aragon, and D. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. 18, 497–506 (2001). 14. A. Cornejo-Rodriguez, “Ronchi test,” in Optical Shop Testing D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 317–350. 15. R. M. Clare and R. G. Lane, “Wave-front sensing from a subdivision of the focal plane with a lenslet array,” J. Opt. Soc. Am. A 22, 117–125 (2005). 16. J. M. López-Ramírez, D. Malacara-Doblado, and D. MalacaraHernández, “New simple geometrical test for aspheric lenses and mirrors,” Opt. Eng. 39, 2143–2148 (2000). 17. D. Malacara-Doblado and I. Ghozeil, “Hartmann, HartmannShack and other screen tests,” in Optical Shop Testing, D. Malacara, ed., 3rd ed. (Wiley, 2007), pp. 361–394. 18. D. P. Salas-Peimbert, G. Trujillo-Schiafino, D. MalacaraHernández, D. Malacara-Doblado, and S. Almazán-Cuellar,
19. 20. 21. 22.
23. 24.
“Ophthalmic lenses measurement using Hartmann test,” Proc. SPIE 5622, 102–106 (2004). Y. Mejia-Barbosa and D. Malacara-Hernández, “Object surface for applying a modified Hartmann test to measure corneal topography,” Appl. Opt. 40, 5778–5786 (2001). R. Díaz-Uribe and M. Campos-García, “Null screen testing of fast convex aspheric surfaces,” Appl. Opt. 39, 2670–2677 (2000). P. Su, R. E. Parks, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49, 4404–4412 (2010). P. Su, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field x-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20, 12393– 12406 (2012). T. Iwane, “Automatic lens meter,” U.S. Patent No. 4,779,979 (25 October 1988). T. Iwane, “Automatic lens meter,” U.S. Patent No. 5,349,433 (20 September 1994).
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In this paper we will review some of the many different practical arrangements that have been obtained to measure the transversal aberrations of optical systems based on the odd and vulnerable Hartmann test. There are many optical testing configurations that apparently are not related to the original Hartmann test. However, they are really the same thing and can be considered just a variation of the same basic arrangement, as will be described here. © 2015 Optical Society of America OCIS codes: (260.0260) Physical optics; (120.0120) Instrumentation, measurement, and metrology; (120.3930) Metrological instrumentation. http://dx.doi.org/10.1364/AO.54.002296
1. Introduction
The Hartmann test was invented by Hartmann in Germany more than a century ago [1]. Surprisingly, it is still in wide use today. The reason is its great simplicity and the sensitivity it has for measuring optical systems of many kinds. What we would like to point out here is that there are many optical testing configurations that apparently are not related to the original Hartmann test, but they are really the same thing and can be considered just a variation of the same basic arrangement. The mathematical analysis is the same. They have in common that the experimental measurements are the slopes of the surface or wavefront under measurement. In many systems this is equivalent to measuring the transverse aberration. 2. Measurements of the Transverse Aberrations and Integration to Retrieve the Wavefront
Let us consider a wavefront with an almost spherical shape, which can be aspheric. For example a plane wavefront is just an spherical wavefront with an infinite radius of curvature. The relation between the wavefront slope’s transverse aberrations and the wavefront deformations is surprisingly simple and 1559-128X/15/092296-06$15.00/0 © 2015 Optical Society of America 2296
APPLIED OPTICS / Vol. 54, No. 9 / 20 March 2015
accurate (see Fig. 1). The exact relation was described by Rayces [2]: ∂Wx; y and ∂x ∂Wx; y TAy −r − Wx; y ; ∂y
TAx −r − Wx; y
(1)
but the wavefront deformations are so small compared with the radius of curvature of the wavefront that these expressions can be written as TAx −r
∂Wx; y ∂x
and TAy −r
∂Wx; y : ∂y
(2)
The integration methods for these two expressions in order to retrieve the wavefronts are many and have been the subject of research for more than a hundred years. All of these methods can broadly be classified as zonal or modal methods. The modal methods do not integrate the positions of the spots, as determined by their centroids, one by one as do the zonal methods, like that by Ghozeil and Simmons [3] or by Southwell [4]; instead, they integrate the pattern with the spots as a whole, assuming that the wavefront deformation can be represented by a combination of linearly independent analytical functions. These functions can be called aberrations and can
The selection of the wavefront analysis, modal or zonal, is quite important and depends on the type of wavefront aberration that is to be measured. If we expect to find small extent irregularities and low order aberrations, a zonal procedure is convenient. On the other hand, if high order polynomial aberrations are to be detected, a modal processing is more convenient. In some cases it may be necessary to analyze the wavefront with both methods. The detailed procedure should also take into account the pupil shape or any possible central obscuration. 3. Different Configurations Fig. 1. Relation between the wavefront deformations and the transverse aberrations or wavefront slopes.
Fig. 2. Classical Hartmann arrangement proposed in 1900.
have many different representations. Frequently, Zernike polynomials are used in the modal methods to represent the wavefront deformations. Modal methods started with the seminal work by Cubalchini [5] and then were described by Gavrielides [6] and later by Aksenov and Isaev [7], by Acosta et al. [8], and by Prieto et al. [9]. Another example of modal methods is based on Fourier techniques, as shown by Freischlad and Koliopoulos [10] and Canovas and Ribak [11], which can avoid the need for the centroid determination.
The Hartmann test has been so successful that many different configurations had been devised to measure wavefront or surface slopes and from those to retrieve the wavefront deformations. The classical arrangement, proposed by Hartmann in 1900 [1], is shown in Fig. 2, where the mirror or lens under test is covered with a screen with holes, and the transverse aberrations are measured by the deviations of the spots generated by each hole on the Hartmann pattern. Thus, the sampling plane is at the exit pupil of the system or at the mirror, and the observing plane is near, but not exactly at, the focus, so the spots are defocused. For this reason the measured positions are the centroids of the spots. If the mirror is spherical and the light source is close to the center of curvature of the mirror, the spot array at the observation plane is identical to the holes’ square array at the mirror, only scaled down. Hence, if the Hartmann screen is a square array of holes, the spots in the Hartmann plate also form a square array. If the mirror is an ellipsoid and the light source is at one of the foci we also have a square array at the Hartmann plate if the Hartmann screen is a square array of holes. We say that we have a null test if the array of spots at the Hartmann plate is a perfect square array of spots, when there is no undesired aberration at the optical system. If the mirror is aspheric and the light source is at the paraxial center of curvature of the mirror, the spots are along curved lines. This is not a null test. However, we may have a null test if the holes at the Hartmann screen have a distribution such that the array of spots at the
Fig. 3. Shack–Hartmann test. (a) Arrangement proposed by Platt and Shack in 1971 and (b) testing of collimated wavefronts. 20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS
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Fig. 4. Modified Hartmann test, analogous to a Ronchi test.
Hartmann plate is perfectly square when the mirror is a perfect aspherical surface. An extremely interesting and useful modification was proposed by Platt and Shack [12] in 1971, in order to measure collimated (flat) wavefronts, as in Fig. 3. With this arrangement we can measure collimated wavefronts in lenses or telescope mirrors, or even the wavefront aberrations of the human eye, as described by many researchers, for example by Hoffer et al. [13]. The main differences with the basic Hartmann test are that a collimated wavefront with a small diameter can be measured and that the spots are focused and their diameters are determined by diffraction. If the Shack–Hartmann lens array is a square array, as in most cases, we have a null test only for flat or slightly spherical wavefronts.
The Ronchi test [14], as the Hartmann test, measures the transverse aberrations, but only in one direction, perpendicular to the lines in the Ronchi ruling. If the ruling is substituted by a dark screen with a rectangular array of holes, as in Fig. 4, the arrangement can be considered a modified Ronchi test, but also a modified Hartmann test. We may also think in the superposition of two lateral shear interferometers, with shears in two orthogonal directions. If the magnitude of the lateral shear is small, conceptually this test is identical to a Ronchi test with two crossed rulings or to another modification of the Hartmann test. An important difference with the classic Hartmann test is that the sampling and observing planes are interchanged. An observing eye or camera has to be used to observe the virtual images of the spots located at the exit pupil or at the mirror. An interesting application of the same principle has been reported by Clare and Lane [15]. Null test configurations are like in the classic Hartmann test. Another possible configuration is by placing a small aperture imaging lens at the point conjugate to the virtual object position, as in Fig. 5. In front of the lens, near the front focal plane, an array of light sources is located. This array of light sources can be a LED array or even a nonluminous object with a reticle drawn on it. This arrangement can be imagined as a modification of the previous one, based on the Ronchi test, where the sampling plane (the array of light sources) is at the front of the entrance pupil of the system and not at the back of the system, after the exit pupil. The observing eye (Ronchi ruling) or camera should focus on the lens
Fig. 5. Testing with an array of small light sources.
Fig. 6. Testing of a lens with a screen with a rectangular array of lines. (a) Aspheric lens and (b) lens with spherical aberration. 2298
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or on the exit pupil of the optical system under test. Then, as a consequence, the spots are not focused on the measuring plane, as in the original Hartmann test. The small imaging lens and the virtual object are at conjugate positions, where the lens is assumed to be working. As before, the array of light sources can be replaced by a screen with a rectangular array of lines, and the imaging lens by the naked eye as in the testing of a lens in Fig. 6, as described by López-Ramírez et al. [16]. This is a useful tool for aspheric lenses. A null
test configuration is obtained when the virtual object position and the imaging lens are at the proper aberration-free conjugate positions for which the lens was designed. Other possible modified arrangements for null tests of concave paraboloidal mirrors or for concave spherical mirrors with ruled nonluminous screens are illustrated in Fig. 7 and described by Malacara-Doblado and Ghozeil [17]. These previous configurations can have several variations; for example, the lens under test does not necessarily have to be a convergent optical system. A divergent lens, for example an ophthalmic lens, as shown by Salas-Peimbert et al. [18], can be examined as in Fig. 8(a). The imaging system or observing eye should focus on the lens (or exit pupil) and not the light sources, so the spots are not necessarily in focus. This is convenient in order to associate each spot to its corresponding lens position. A flat glass window can be examined as in Fig. 8(b). Car glass windshields windows can be evaluated with this arrangement. The observing eye or camera should be focused on the glasses window under test, producing slightly defocused spots on the observing plane. However, a large error is not introduced if the array of light sources is focused instead of the glass window. With this configuration all the images of the light sources on the ellipsoid are in a plane, very close to the strong convex surface being measured. Since the diameter of the convex surface is small compared with the length of the ovoid, the positions of the virtual image of the light sources can be considered as being at this convex surface. An additional advantage of this configuration is that the spots are focused
Fig. 8. (a) Testing a divergent lens and (b) testing a glass window with a rectangular array of light sources.
Fig. 9. (a) Testing of a convex reflecting surface or a convex mirror with an array of light sources and (b) testing a convex mirror with a cylindrical ruled screen.
Fig. 7. Arrangements to test (a) a paraboloid and (b) a sphere with a ruled screen.
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Fig. 10. Testing with an array of light sources, placing their images (spots) all at approximately the same position.
on the detector as the observing plane. A strong convex surface, for example the corneal surface of the eye, can be measured as in Fig. 9(a). If the virtual observation surface with the images of the light sources is desired to be a flat surface, the array of light sources has to be over an ovoidal surface, as shown by Mejia-Barbosa and Malacara-Hernández [19]. This configuration has been used to measure the corneal topography of the eye. Instead of the ellipsoidal surface with the light sources, a cylindrical surface with the light sources or a dark screen with an array of lines can be used, as shown in Fig. 9(b). The spots in the image are not exactly in a plane, and not all the sampling surface with the light sources or lines array is in perfect focus. But since the center of gravity of the spots does not move, the accuracy of the instrument is not seriously affected as shown by Díaz-Uribe and Campos-García [20]. The array of light sources or of lines cannot form a square array if a null test is desired. The proper array configuration to be used should be calculated by ray tracing, in order to obtain a square array and hence a null test. A similar arrangement with the line arrays on a flat surface or flat computer display has been used with success with the name SCOTS to test telescope mirrors by Su et al. [21,22]. Another arrangement, where the spots are all at the same position, can be devised. The advantage is that the detector can be much smaller, increasing the sensitivity of the measurements. However, since the spots are superimposed at a single position, the light sources are turned on one by one and not all at the same time, as in Fig. 10. This system has been used in some commercial lens meters [23,24]. The focusing lens has the purpose that the cones of light arriving to the lens to be measured are all parallel to the optical axis in order to measure its back focal length. The position sensing detector is conjugate to the collimator and stop in front of the array of light sources. 4. Some Comments about the Transverse Aberration Measurements and the Reference Wavefront
The transverse aberrations are the deviations of the centroids of the spots with respect to their reference positions, with components TAx and TAy . After integration, the results represent the wavefront deformations with respect to the reference wavefront. There are several possibilities for the reference positions for the spots: (a) When the position for the reference spots is a single point at the optical axis. Then the retrieved 2300
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wavefront represents the deformations with respect to a reference sphere with a different curvature than that of the wavefront, with center of curvature at the defocused observing plane. (b) If we have a null test, that is, if the unaberrated system produces a perfect square array of spots, and the position for the reference spots is obtained by generating a perfect array of spots identical to the sampling screen but scaled to a size that fits as much as possible to the measured array of spots, using a least squares procedure. Then the reference wavefront is a sphere with a curvature nearly equal to that of the wavefront. (c) If we have a nonnull test, the positions for the reference spots have to be calculated by ray tracing through the ideal optical system. Then, the retrieved wavefront represents the deformations with respect to the ideal wavefront, which could be spherical or aspherical. 5. Conclusions
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