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A new method to design Hartmann type null screens to test either qualitatively or quantitatively fast plano-convex aspherical lenses is presented. We design both radial and square null screens that produce arrays of circular spots uniformly distributed at predefined planes, considering that the CCD sensor is solely placed inside the caustic region. The designs of these null screens are based on knowledge of the caustic by refraction and on exact ray tracing. The null screens also serve to improve the alignment in optical systems.
© 2016 Optical Society of America
1. Introduction
Aspheric lenses are widely used in both imaging and non-imaging optical systems, because this kind of lens eliminates spherical aberration and reduces other aberrations. These can yield sharper images more efficiently than spherical lenses do. In other words aspheric lenses can help simplify optical system design by minimizing the number of elements required, they are currently useful for correcting distortion in wide angle lenses.
There are several methods to test this kind of lens, in particular Hartmann and Ronchi tests have been a topic of research for many years. In these tests, sampling screens with either holes or fringes with varying separations are used to test lenses and mirror surfaces [1–3]. The Hartmann test traditionally uses a screen with a uniform distribution of holes, which produces a nonuniform distribution of bright spots at the detection plane either by reflection or refraction. For fast optical surfaces, i. e., F/# < 1, this non-uniformity can be very pronounced, complicating the subsequent analysis. To overcome this problem null tests have been developed for aspherical convex surfaces [4] and off-axis mirrors [5], in which instead of using an array of point light sources, null screens printed with a commercial laser printer are used to mask a uniform light source. Null tests are motivated by the fact that the interpretation of the surface shape is simplified, and the visual analysis turns out to be straightforward for both qualitative and quantitative tests.
In earlier papers, special rulings with curved lines were constructed to test aspherical concave mirrors and this approach was denominated the null Ronchi test. In [6] the ruling was computed according to the following procedure: using a ray tracing program, a transverse aberration curve (TA) is computed at the approximate place where the ruling is to be placed. A system of five linear simultaneous equations is numerically solved in order to determine the coefficient values of the aberration polynomial TA. Finally, using geometrical considerations and the computed TA the null screens are designed. Alternatively, in [7], a predetermined formula is used for the TA to compensate for the error introduced by the Ronchigram fringes over the mirror surface, and going through geometrical considerations, a simple formula is obtained and solved by using Newton’s method to design null screens. At the present time it is possible to design null screens using commercial lens design programs [8,9]. Alternatively, a method to design Ronchi-Hartmann screens that produce aligned straight fringes for predefined planes of detection near to the focal distance of the lens under test, based on knowledge of the caustic by refraction on spherical lenses was presented in [10]. It is important to state that in all references mentioned above [1–10] the null screens are designed in such a way that the plane of detection will be placed outside of the caustic region.
The Hartmann test measure the wavefront slopes, and they are directly related to the ray transverse aberrations, furthermore, as is well known, there are several integration methods to obtain the wavefront deformations propagated through lenses under test [11–14]. The simplest Hartmann screen is one with only four holes placed within a circle, separated by an angle of 90 deg, which have been properly implemented to align and quantitatively test optical systems [15].
Recently, an attempt to describe the evolution of the null Ronchi-grating considering the plane of detection to be placed inside the caustic region, based on the caustic touching theorem was presented in [16, 17], and references cited there in. Unfortunately the null Ronchi-grating obtained in these references does not properly work experimentally because these null screens are not binarized. In other words, the authors have just projected straight lines instead of strips or bands to be recorded at detection plane, and therefore the Ronchi-grating does not have a predefined area. Consequently the lines projected do not present black and white stripes uniformly illuminated, nor do they have equal spacing between contiguos fringes, as is usually done in the null screen method. In this paper we design a null screen that produces a uniform distribution of spots in which the interspacing between contiguous circular spots have the same width at the detection plane. Due to the position at which the null screen is placed, the procedure can be considered a modified Hartmann test [18]. The design of this screen is based on knowledge of the caustic surface by refraction also called diacaustic and the exact ray tracing, and does not require the aberration polynomial TA as the references above mentioned do. As is well known, the testing of an optical system is best accomplished if no additional optics are needed. An important feature of this paper is that we use only a CCD sensor at the detection plane inside the caustic region without any additional lenses to focus the Hartmanngram to be recorded at the CCD sensor. It is worth noting that this optical test is in the geometrical optics regime; therefore the effects of diffraction are not considered here, and further work should consider these issues.
2. Theory
Throughout this manuscript we define the Z axis parallel to the optical axis, we assume that Y − Z is the plane of incidence, which is a cross section of a plano-convex aspheric lens whose paraxial radius is R, and the origin of the system 𝒪 is placed at the first vertex of the lens. We assume that there is rotational symmetry about Z axis and a plane wavefront impinging on the left side of the lens, crossing the plane face of the lens without being deflected, and it is propagated to the aspheric surface where will be refracted outside of the lens. Let H be the entrance aperture, t the thickness, ni the index of refraction of the lens for a predefined wavelength, considering that the lens is immersed in a medium with index of refraction na, normally in air, therefore ni > na, and where we have assumed that ShN represents the aspheric equation in a meridional plane written as
where c = 1/R is the paraxial curvature, k is the conic constant, A2, A4, . . . , A2N, are the aspheric coefficients with N the number of aspheric terms included in the polynomial and h represents the height for each one of the arbitrary incident rays, where Eq. (1) is valid in the range h ∈ [−H, H]. It is important to state that if N = 0, then Eq. (1) is reduced to the equation which represent the profile for conic surfaces. According to [19] the tangential caustic surface for a plano-convex aspheric lens when a plane wavefront is propagated along the optical axis and refracted outside of the lens can be parametrically written as where the subscript d means diacaustic and S′hN, S″hN are the first and second derivative with respect to h from Eq. (1) respectively. Alternatively, an equation to represent all refracted rays outside of the aspheric lens, according to [19] was expressed as It is important to remark that Eq. (2) gives the coordinates of the locus of points that parametrically represent the envelope of the family of refracted rays produced by an aspheric lens in a meridional plane provided by Eq. (3), where z0 is an arbitrary distance along the optical axis, and y0 its respective distance along the Y-axis for an incident ray coming from a unique height hs for z0 ≥ f as is shown in Fig. 1(a), where the subscript s means screen. We can clearly see from Eq. (3) that y0 is a function of h and z0, where we have assumed all the parameters of the lens involved in the process of refraction are constant. From Eqs. (2) and (3) if the radical , imposes the condition for Total Internal Reflection (TIR), therefore, the entrance aperture in this configuration is naturally limited by this condition from H to hc, where the subscript c means critical angle, in other words, for h > |hc| the rays undergo TIR, and therefore they are not refracted outside of the lens. Alternatively, rewriting Eq. (3) we obtain a polynomial equation for h given by this equation can be solved through numerical methods to make straightforward the design of null screens instead of using commercial lens design programs. Without loss of generality we could remove the value for t from Eq. (4), mathematically it is possible if we assume a translation for the origin of coordinates from 𝒪 to t as is shown in Fig. 1(a), where the numerical value for hs remains unalterable under this mathematical transformation. Throughout this manuscript we use the following parameters for the aspheric lens under test: it possess three aspheric terms N = 3, whose values are: A2 = 7.9 × 10−6, A4 = 1.5 × 10−7, A6 = 1.3 × 10−9, with R = 13.8595mm, t = 11.04mm, k = −1.0, na = 1, considering ni (B270) = 1.5212 for λ = 633nm, its diameter is D = 30mm, EFL = 26.5mm, and an entrance aperture H = ±15mm, which is limited to h ∈ [−hc, hc] with hc = ±12.04mm. We also use a rectangular ccd sensor whose major and minor lengths are LM = 8.8mm and lm = 6.6mm respectively, with a resolution 640 × 480 pixels, obtaining a square pixel whose linear length is Lp = 0.01375mm, where the subscript p means pixel. The light source used is a He-Ne laser (λ = 633nm). The diagram of the experimental setup is shown in Fig. 1(b), where we can also see two polarizers near laser beam, which have exclusively been used to reduce the amount of irradiance impinging on the CCD sensor, and we have also implemented a collimator lens with a large focal distance in order to reduce a hot spot exposed in the central part at CCD sensor. An important consideration is that if visual observations are to be made in real time, the non-uniform holes used as null screen can be calculated to produce an observable pattern of adequate size for the lens under test at a unique detection plane.
Fig. 1 (a) Caustic produced by refraction through a plano-convex aspheric lens and their associated parameters considering that the point source is located at infinity. (b) Diagram of the experimental setup to test a plano-convex aspherical lens considering a ccd sensor placed inside the caustic surface.
3. Circular spots uniformly arranged at detection plane to design the null screens
To design null screens, we follow all the steps explained in [10,20–22], where square and linear arrays of circular spots uniformly spaced in the plane of detection have been designed, which have been placed outside of the caustic region. In other words, by solving Eq. (4) through numerical methods, yields a unique solution for each point belonging to either circular or linear patterns, providing either non-uniform drop spots or irregular curves respectively which allow us to retrieve a uniform array pattern with similar sizes recorded at detection plane, only if the surface under test does not have any deformation on their refracting surfaces. The main objective is to fill with circular spots as much as possible a circle inscribed in a square area belonging to the sensor which has a rectangular active area provided by (LM × lm) mm2, being LM and lm the major and minor lengths respectively of the CCD sensor. For simplicity we have chosen both square and quasi-angular arrays as are shown in Figs. 2(a) and 2(b), to be drawn at detection plane such that y0 ∈ [−lm/2, lm/2]. Furthermore, we should regard for practical purposes the size and technical properties of the sensor, taking into account that the number of spots already defined to be registered inside the active area of the sensor. For the null screens whose images are recorded near the focal distance, they very often produce a central hot spot, which makes it difficult to process the Hartmanngram in order to evaluate quantitatively the centroids for all spots images.
Fig. 2 (a) Square array of circular spots at detection plane to design null screens with P = 13, n = 113 spots, lm = 6.6mm and Δ = 0.264mm. (b) Distribution of quasi-angular array of circular spots at detection plane to design the null screen with n = 133 spots.
We briefly explain the process of designing null screens, which will be placed at the entrance aperture of the lens under test, in such a way that we get images recorded uniformly and well distributed within CCD sensor, and whose holes should cover as much area of the entrance aperture of the lens under test as is possible. It is enough to describe entirely the process to obtain a square array of spots inscribed in a circumference inside the ccd sensor, although the process could be extended to draw arbitrary patterns at detection plane. To calculate the heights hs of the points on the null screen that yield non-uniform holes, we proceed backwards, starting at detection plane Z = z0. We predefine a priori a number of circular spots as P × P, where P is the number of circles along X and Y-axis with a uniform interspacing between bright spots defined by Δx and Δy, demanding simultaneously to get bright spots at the center and also at borders of the sensor along the Y-axis, therefore P must be an odd integer number. It is easy to show that the equations of those circles can be written as
where the interspacing and the diameter of the bright spots is defined by Δ = lm/(2P −1) = 2rs, and the coordinates of the centers can be written as In order to supply the length y0, we consider an appropriate transformation of coordinates correlating the image plane inside a meridional plane. We can see that the distance y0, is provided if we intersect straight lines; y = (tan α)x with the circumferences given in Eq. (5) as was considered in [20], where α is a variable slope to be intersected along with all the points around of circular spots. Alternatively, we can parametrize the circumferences of Eq. (5) as follows all the circles will be formed as a set of M discrete points, which form part of a continuous curve joined with straight lines. In our particular case we consider an angular separation between contiguous points given by θM = 2π/M, yielding a regular polygon with M sides inscribed in a circumference with radius rs. Furthermore, if the number M is very large, the contiguous points are not too widely separated, then the polygon turns out practically to be a circumference. Thus, the distance y0 can simply be written as , in such a way that we restrict the area of the spots placed inside of a circular region at CCD sensor, reducing the number of bright spots, for instance, P = 9, yields n9 = 49 spots, for P = 11 yields n11 = 81 spots, for P = 13 we get n13 = 113 spots and so on. Finally, to properly design the Hartmann type null screens having Q holes, we must take into account the rotation’s angle for y0 provided in the plane of detection, given by φij = arctan (yjq/xiq), as are shown in Fig. 2(a), and for a quasi-angular array in Fig. 2(b). For convenience we have chosen the radius length “rs” to cover a few pixels facilitating the printing of the null screens on foil sheets and also for the processing of the images recorded at CCD sensor. We substituted all the parameters of the lens provided above into Eq. (4), in order to design a suitable null screen for a lens under test considering the following three regions. (1) Placing the plane of detection along the optical axis ranged at t ≤ z0 < zm, where zm is the intersection formed between lower and upper marginal rays refracted through the lens, we have a unique solution for hs, whose values lie in the range hs ∈ [0, hc] for y0 > 0 and hs ∈ [−hc, 0] for y0 < 0, where hc is the critical height of the lens under test, this region we have called Region I. (2) Alternatively, if detection’s plane is placed near to z0 = zm, there are two possible solutions for hs, only if y0 lies under the curve of the caustic surface, obtaining both solutions of hs ∈ (0, hc) for y0 > 0 and hs ∈ (0, −hc) for y0 < 0. In other words, there are rays propagating through the central part hs1 and simultaneously around the borders of the lens hs2 and they are brought in coincidence at y0 > 0, this region we have called Region II. (3) Finally, if the detector is placed at zm < z0 < 1.0 EFL the plane of detection completely lies inside the caustic surface, in this particular case there are three possible solutions for hs because we have rays from the lower part hs1 < 0 and rays arriving from the upper part of the lens hs2 > 0 and hs3 > 0 which coincide at the point y0 > 0, only if the y0 lies under the curve of the caustic surface, this region is called Region III, as is shown in Fig. 3(b).Fig. 3 (a) Three different polygons with M sides forming part of the null screens, which will be overlapping at plane z0, the points whose distance are hs1, hs2 and hs3 are brought in coincidence at the point y0. (b) Distribution of the regions along the optical axis showing the coincidence of three refracted rays at the point (z0, y0). (c) Parameters involved in the process to design a circular bright spot at detection’s plane.
If we place the sensor at z0 = CLC, where CLC means Circle of Least Confusion, we increase the entrance aperture of the lens under test, additionally at this plane there are three possible solutions for y0 > 0, which are represented by hs1, hs2 and hs3 as are shown in Fig. 3(a). Finally, to design the Hartmann type null screens after we have obtained numerically the values for hs from Eq. (4) we must apply the transformation given by
where φij = arctan (yjq/xiq) as is shown in Fig. 3(c), selecting two contiguous points and joining them through straight lines, in order to close a quasi-continuous curve formed for a polygon with M sides, the inner area for all Q polygons will be transparent and the outer area will be blackened by using a commercial laser printer, in such a way that all Q polygons will be transformed in Q holes forming the null screens, which yield n bright spots at detection plane. On the other hand, the null screens could be printed on an acetate foil using a traditional laser printer specified at 2400 dpi, although they can be displayed by using a Spatial Light Modulator (SLM), or alternatively they could be printed by photographic or lithographic process.4. Null Screens: Square and quasi-angular arrays
By substituting all the parameters of the lens under test provided above into Eq. (4), by using Eqs. (6) and (7), where we have considered P = 13 spots along Y-axis, with M = 360 points for each circular spot, and placing the CCD sensor at different planes along the optical axis for z0 = t, 15, 19, 23, 27, 31, 34 and 1.0EFL mm, as are shown in Fig. 4(a), we assume that the CCD sensor should record an ideal Hartmanngram either square or quasi-angular bright spots arrays as are shown in Figs. 4(b) and 4(c) respectively. For a square array we get n = 113 bright spots and for a quasi-angular array we get n = 133 bright spots, in a such way that the number of bright spots inside the sensor for a quasi-angular array is greater than for a square array. Following the steps explained above we obtain the points hs, whose values are substituted into Eq. (8), yielding distinct null screens to be recorded at different planes z0 for square arrays as are shown in Fig. (5), and for quasi-angular arrays as are shown in Fig. (6). We have included a video showing the evolution of the null screens moving the planes of observation along the optical axis. We clearly see in both Figs. (5) and (6), that placing the CCD at z0 = t mm, both square and quasi-angular null screens are practically equal to the original spot arrays displayed in Figs. 4(b) and 4(c) respectively. Alternatively, placing the CCD in the range t < z0 ≤ 22mm, the sensor completely lies inside the caustic region, and the central part of the null screens increase their sizes. Finally for z0 > 22mm, the outer part of the CCD sensor lies outside of the caustic region, producing a ring-shaped hole, which completely contains the border of the caustic surface, and this ring-shape hole will collapse into a circular area near to z0 = 1.0 ELF. Additionally, we can see that all the null screens are formed with drop spots having non-uniform areas. Furthermore, for z0 ≤ 22mm the outer drop spots formed by the solutions hs1 are flatter than inner drop spots formed by the solutions hs2, only if the sensor lies inside the caustic region as is shown in Fig. 3(a), on the other hand, if the sensor lies outside of the caustic the holes resemble like drop spots. The main idea is that the CCD sensor completely lies inside the caustic region, because the irradiance recorded at the sensor will be quasi-uniform, consequently making straightforward the process to obtain the centroids of the spots using the images recorded on the CCD to evaluate qualitatively the lens under test. Therefore we have chosen that the sensor will be placed at z0 = 22mm, for both square and quasi-angular arrays, whose null screens are shown in Fig. 7 respectively, considering the values for P = 9, 11 and 13, and also an exact ray tracing showing a cumulative distribution of rays at borders of the lens and a uniform distribution of rays impinging on the CCD sensor placed inside the caustic region.
Fig. 4 (a) Designing null screens placing the ccd sensor at different planes of detection. (b) Zoom of a square array of bright spots on CCD sensor to design the null screens. (C) Zoom of a quasi-angular array of bright points on CCD sensor to design the null screens.
Fig. 5 Design of square null screens where Q is the number of holes, whose images are recorded at different planes of detection z0, for all cases supplying n = 113 bright spots (see Visualization 1).
Fig. 6 Design of radial null screens where Q is the number of holes, whose images are recorded at different planes of detection z0, for all cases supplying n = 133 bright spots (see Visualization 1).
Fig. 7 Design of square and quasi-angular null screens to be recorded at z0 = 22 mm for P = 9, 11 and 13 drop spots along Y-axis and its ray tracing process on a meridional plane.
There are slight non-uniformities of intensity recorded on the CCD sensor, which could be produced by several factors. For instance there is an inevitable non-isotropy of light exiting from the collimator lens producing a quasi-monochromatic and quasi-plane wavefront propagating along the experimental setup, which could modify the images recorded.
Additionally, we have printed the null screens on acetate foil, assuming that the acetate foil is a perfect plane-parallel sheet and that all foils are free of strain and stress after the printing process, furthermore, we are assuming that the printer produces a perfect mark impressed on the foil surface, these imperfections can considerably alter the images recorded on the sensor. Finally, possible slight misalignments in the experimental setup could also modify the images recorded, these insignificant non-uniformities are shown in Fig. 8(a) for square and Fig. 8(b) quasi-angular null screens respectively, even more in all figures the central part resemble to have a kind of coma aberration.
Fig. 8 Images recorded at ccd sensor for (a) square and (b) quasi-angular arrays null screens considering three different holes being illuminated by a plane wavefront and brought in coincidence at detection plane, which is placed at CLC.
Alternatively, if we exclusively remove the external flattest spots for all null screens provided in Fig. 7, it is possible to obtain better images than those recorded in Fig. 8. The amount of area covered by using these modified null screens on lenses under test is not substantially reduced. There are two ways to remove the external spots, the first is consider do not print the flattest spots in the null screens, the second method is to place an iris in front of the lens under test and reduce slightly the entrance aperture to improve the images recorded at CCD sensor as is shown in Fig. 9. For all square arrays recorded within sensor the external circular spots resemble a slightly decentered pincushion aberration, as is shown in Fig. 9(a). For all quasi-angular arrays recorded on the CCD sensor the external circular spots resemble a slightly astigmatic aberration as is shown in Fig. 9(b).
Fig. 9 Images recorded at ccd sensor reducing slightly the illuminance at entrance aperture for (a) square and (b) quasi-angular arrays null screens being illuminated by a plane wavefront and brought in coincidence at detection plane z0 = 22mm.
It is important to remark that the CCD sensor has a frontal glass substrate whose technical properties are unknown, in other words, neither thickness nor refractive index for this substrate are provided by the manufacturer, and this substrate entirely covers the charge couple device, which alters the final images recorded on this device. An error analysis and a quantitative test for measuring the thickness and index of refraction for this substrate must be developed in order to improve this test. We are planning in future works the implementation of a CCD sensor with higher resolution and without any glass substrates or thin films covering the charge couple device area. The principal idea of this paper is to extend this result to obtain the shape of the lens under test by using null screens through a modified procedure of the Hartmann test [23]. The surface departures from the best surface fit are of the order of one μm when the errors in the determination of the coordinates of the centroids of the refracted images are less than 1 pixel, and the errors in the coordinates of the spots of the null screen are less than 0.5mm, according to reference [5]. Furthermore, an error analysis including the numerical precision for each hs and their positioning after the printing on the acetate foil are necessary in order to improve the precision of the null Ronchi-Hartmann test for a quantitative evaluation.
The test is very sensitive to alignment and positioning. For example in Figs. 10(a)–10(c) and 10(e)–10(g) the CCD sensor has been slightly displaced toward the lens under test considering square and quasi-angular arrays of bright spots, we can clearly see that the bright spots are completely disordered, although the null screen is well centered, which means that it is aligned with the optical axis producing symmetrical patterns. In Figs. 10(d) and 10(h) there are misalignments of the null screens, in other words, the null screen has been laterally displaced along the X-axis, and the loops are asymmetrical when the experimental setup is not centered. Finally, in Figs. 10(i)–10(l) the recorded images were obtained considering that the CCD sensor was placed slightly outward of the lens under test, and they resembles a zoom of the center of the null screen. As was explained above, we can provide the values for a regular grid in the observation plane in the form of a square array, in a similar way as Ronchigrams does, and obtain the shape of the lens under test in future works.
Fig. 10 (a)–(c) and (e)–(f) CCD sensor has been placed slightly toward the lens under test for square and quasi-angular arrays respectively. (d) and (h) the null screen has been misalignments. (i)–(l) CCD sensor has been placed slightly outward of the lens under test for square and quasi-angular arrays respectively.
5. Conclusions
A simple method to design null screens in order to test a fast plano-convex aspheric lens has been presented. We have designed Hartmann type null screens considering that the CCD sensor is solely placed inside the caustic region, as a consequence this test notably provides uniform bright spots recorded at the CCD sensor. The test is very sensitive to positioning and alignment of optical systems, as we can see from the images recorded inside the caustic region. Although the polynomial Eq. (4) can’t be solved analytically it is enough to solve it through numerical methods. This method opens the door to designing null screens to test in both configurations either convex-plano or plano-convex aspheric lenses either outside or inside of the caustic region. As was explained above, this optical test is in the geometrical optics regime; therefore the effects of diffraction are not considered here, and further work needs to consider these issues.
Funding
Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica – Universidad Nacional Autónoma de México (PAPIIT-UNAM) (IN112316, IT101216); Consejo Nacional de Ciencia y Tecnología (CONACyT) (168570); Escuela Superior de Ingeniera Mecánica y Eléctrica, Unidad Ticomán, Instituto Politécnico Nacional (20161277).
Acknowledgments
GCS and DCR wants to give special thanks for the scholarship granted by CONACyT. The corresponding author is grateful to I. Goméz-García for their valuable assistance and comments.
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