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Abstract
We have implemented an exact ray trace through a plano-freeform surface for an incident plane wavefront. We obtain two caustic surfaces and provide the critical points related to the ray tracing process. Additionally, we study the propagation of the refracted wavefronts through the plane-curved surface. Finally, by using the Ronchi-Hartmann type null screen and placing the detection plane within the caustic region, we have evaluated the shape of a plano-freeform optical surface under test, obtaining an RMS difference in sagitta value of 6.3 μm.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nowadays most optical systems are rotationally symmetric, because such systems are the simplest to deal with, and there exist many methods for testing their shape and performance with either geometrical or interferometrical tests. It is common to design non-rotational symmetric systems, with properties which significantly differ from rotational symmetry optics. In particular, those elements are called freeform optical surfaces, and they provide a third independent axis during the fabrication process to create an optical surface having non-symmetric features [1]. This capability of adaptation allows designers to increase the fields of usage to areas such as imaging and non-imaging forming optics, providing more complex optical systems which have diversified to achieve applications in multidisciplinary approaches. Therefore, it is essential to verify if the instrumentation traditionally implemented to quantitatively evaluate the properties of these new optical systems is still valid.
Additionally, a freeform surface can be mathematically expressed by equations with two variables, or by small segments of continuous equations, such as splines, $B$-splines [2]. These have been utilized in imaging and non-imaging optical systems, they can be designed either for correcting aberrations or producing uniform intensity patterns with a specific geometric shape [3]. Some techniques to manufacture freeform surfaces consider the use of predefined molds, 3D printers or computer numerical control machines [4]. Alternatively, in the literature there exist several definitions for caustic surfaces, related to their physical or geometrical approaches, each with at least one method to obtain a mathematical representation [5–7]. On the other hand, a set of points where the intensity of the refracted rays becomes infinite [8]. Additionally, considering a three-dimensional transformation between object-image space, we can calculate its critical points [9–11] obtaining the caustic surfaces. In this manuscript we study the formation of caustic surfaces formed through plano-freeform optical surfaces.
Finally, 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 [12–14]. Recently, an attempt to describe the evolution of the null Ronchi-grating considering the plane of detection inside the caustic region was presented in [15,16], and references cited there in. Unfortunately, the null Ronchi-grating obtained in Refs. [15,16], does not properly work experimentally because these null screens are not binarized. Alternatively, there exist papers where null screens have been properly implemented. For instance, for testing plano-convex spherical lens demanding that the detection plane is placed at effective focal length see [17]. Otherwise, for testing aspherical surfaces assuming a detection plane inside the caustic surface see [18]. Finally, a preliminary trial for testing plano-freeform surfaces was presented in [19]. In this paper we design a null screen that produces a uniform distribution of spots, in which the interspacing between contiguous circular spots has the same width at the detection plane, also straight fringes like a Ronchi-grating for testing quantitatively the shape of a Zernike surface were implemented. 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. Preliminaries
In order to implement an exact ray tracing through a plano-freeform refracting surface, we demand to know the mathematical representation of the surface under test, which we have simply called a Zernike surface. For convenience, we assume the equation for the second refracting surface is given with reference to a coordinate system in which the $Z$-axis coincides with the geometrical center of the first face of the optical surface, which is a plane surface and its origin of coordinates is placed at an arbitrary point on $X$-$Y$ plane as is shown in Fig. 1(a). We assume that the rear surface is represented by a continuous and differentiable function with two variables $z=f(x,y)$. Additionally, we demand a bundle of rays propagating parallel to $Z$-axis impinging at the plano surface, passing without deflection through it and these rays are propagated up to the second surface defined by a Zernike polynomial, and finally the rays are refracted outside of the optical surface. In particular, we consider that the second surface can be expressed as an odd Zernike polynomial [20] according to
Fig. 1. (a) Process of refraction in a Zernike surface. (b) Elevation map of Zernike surface with a diameter $D=54.8$ mm. (c) 3D model of the Zernike surface.
Let ${\vec P}_1$ be a point where the incident ray intersects the plano-curved surface whose coordinates are ${\vec P}_1 = (x, y, z_m)$, where $z_m$ is constant, for practical purposes we consider $z_m=0$. Furthermore, the incident ray will be propagated inside the Zernike surface along the direction of a unitary vector ${\hat R}_1= {\hat I}_1$. We also assume that ${\hat I}_1=(0,0,1)$ is a unitary vector representing the direction of the incident ray, and ${\hat N}_1=(0,0,-1)$ is the normal vector on the first refracting surface, the ray is propagated up to the second surface defined by ${\vec P}_2 =\{x_i, y_i, z(x_i, y_i)\}$, where the suffix $i$ means incident point, and $z(x, y)$ was defined by Eq. (2). Following all the steps explained in [7], the refracted ray outside of the Zernike surface is given by
Fig. 2. (a) Exact ray tracing considering a bundle of parallel rays impinging on the Zernike surface on the plane $\theta =75^{\circ }$. (b) Refracted rays projected in a meridional plane $X'$-$Z$ (see Visualization 1). (c) Refracted rays projected in a sagittal plane $Y'$-$Z$, where $X' \bot \,\, Y'$.
For the Visualization 1, from Fig. 2(a), the green continuous curves represent the sagittal caustic surface and the red continuous curves represent the refracted wavefronts. Additionally, we have shown the four axis $X$, $Y$, $X'$ and $Y'$, assuming that a rotation is an anticlockwise rotation through an angle $\theta$ between $X$ and $X'$ axes respectively about $Z$-axis. Let us choose a Cartesian coordinates system $(X, Y , Z)$ in the space under consideration for simplicity in such a way that we can denote a rotation through an angle $\theta$ about the $Z$-axis by ${\mathcal {R}}(\theta )$, and substituting Eq. (3) we get
3. Caustics produced by a Zernike surface
Without loss of generality, we can consider that Eq. (3) provides the direction of propagation for all the rays outside the Zernike surface, which is represented by $\vec {X_2}= (X_{2x},X_{2y},X_{2z})$, and their respective rays are propagated along the unitary refracted vector $\hat {R}_2=(R_{2x},R_{2y},R_{2z})$. In order to obtain the caustic surface produced by the refracted rays through the Zernike surface, we calculate the critical points from Eq. (3), because the singularities of the mapping will define both sagittal and tangential caustic surfaces. Subsequently, we calculate the Jacobian matrix of the transformation according to Eq. (3), obtaining
Fig. 3. (a) Sections of the real and virtual sagittal caustics. (b) Sections of the real and virtual tangential caustic. (c) Sections for real sagittal caustic. (d) Sections for real tangential caustic. (e) Real sagittal and tangential caustics placing an observation plane at $Z=450$ mm.
4. Wavefront produced through the Zernike surface
In this section we study the propagation of the refracted wavefront through the Zernike surface, for a plane wavefront impinging on the refracting surface. In such a way that the initial wavefront reaches the plane face of the Zernike surface, which is connected by the curved interface $f(x,y)$. It is equivalent to require that the optical path length (OPL) of a light ray from a given point on the initial wavefront to the refracting point be equal to a given constant for all the refracted rays passing through the Zernike surface. In other words, all the rays that emerge from the wavefront leaving the Zernike surface should have the same OPL. Therefore, considering an arbitrary vector $\vec {P_1}^{\prime } = \{x, y, z_m \}$, impinging on the Zernike surface, where the rays begin to be refracted outside of the Zernike surface. Thus the OPL is simply determined by the following expression
Fig. 4. Propagation of wavefronts through the Zernike surface, $\mathcal {L}$ represents arbitrary distances, and $\mathcal {L}=0$, is the 0 phase wavefront.
In particular when the parameter $\mathcal {L}$ equate the difference between the minimum $z_m$ and the $z_M$ maximum point on the Zernike surface, we can define the zero-distance phase wavefront ${\mathcal {L}}_0$, in other words, when $\mathcal {L}_{0} = n_l (z_M-z_m)$ the first wavefront that leaves completely of the plane-curved plate is called the zero-distance phase wavefront. Several wavefronts refracted and propagated at arbitrary distances are shown in Fig. 4. It is important to state that traditionally the difference between peak to valley is considerably greater in freeform surfaces than rotationally symmetric systems such as simple lens. Therefore, it is not possible to implement an interferometric test to evaluate the shape of the Zernike surface, for this reason we will implement a geometrical test based on null screens as is explained below in order to quantitatively test the Zernike surface.
5. Design of Ronchi-Hartmann type null screens
To implement the null screens test, we follow all the steps explained in [18], where square and linear arrays of circular spots uniformly spaced in the plane of detection have been designed. In this work we design predefined patterns similar to Hartmann-Ronchi type null screens, corresponding to spots and fringes respectively. Then by solving Eq. (3) through numerical methods for predefined uniform array pattern for either circular spots or straight fringes, we obtain multiple solutions for designing the null screen, which is placed in front of the Zernike surface as shown in Fig. 5. The null screens are formed for either non-uniform spots or irregular curves which allow us to retrieve a uniform array pattern recorded at detection plane, only if the surface under test does not have any deformation. In previous works, a CCD sensor has been used to be placed at detection plane, however, in this work an opaque screen is used at detection plane, because to the area of a Zernike surface is too large to be quantitatively evaluated.
The main idea is to fill with either circular spots or straight lines as much as possible a circle inscribed in a square area belonging to the detection plane provided by $(l_d \times l_d)$ mm$^2$, being $l_d$ the length of the detection plane. For simplicity we have chosen both quasi-angular and linear strips arrays as are shown in Figs. 6(a) and 6(b), respectively, to be drawn at detection plane such that $(X_d , Y_d) \in [-l_d/2, l_d/2]$. Furthermore, we should regard for practical purposes the size and technical properties of the sensor, considering that the number of spots is already defined to be registered inside the detection plane. For the null screens whose images are recorded near the focal distances, they very often produce a hot spots, which make it difficult to process the Hartmanngram in order to evaluate quantitatively the centroids for all spots images as will be show latter. We have chosen the detection plane at $Z_d$ along the optical axis, the predefined pattern has coordinates $( X_d, Y_d )$, to calculate the null screen we must solve for the variables $(x_i, y_i, {\mathcal {L}}_i )$ from the complete set of equations given by
Fig. 6. (a) Predefined pattern of the quasi-angular spots, with $P = 11$ and $Q=95$ circular spots. (b) Ronchi-type vertical stripes predefined pattern, with $Q=9$ bright fringes.
For instance, if we assume $P=1$ then we obtain $I=0$, substituting this value into $\tau _{0}^{0}$ provides a circumference placed at origin. For $P=3$, it yields $I=0,1$, providing one circumference placed at the origin for $\tau _{0}^{0}$ and also a ring with $\tau _1=\pi /3$, yielding six values according to $\tau _{1}^{0} = 0 \, , \, \tau _{1}^{1} = \pi /3 \, , \, \tau _{1}^{2} = 2\pi /3 \, , \, \tau _{1}^{3} = \pi \, , \, \tau _{1}^{4} = 4\pi /3 \, , \, \tau _{1}^{5} = 5\pi /3$. For $P=5$, we get $I=0,1,2$, providing a circumference placed at the origin and two rings, the first has 6 values as was explained above and for the second ring we get $\tau _2 = \pi /6$, obtaining 12 values given by $\tau _{2}^{0} = 0\, , \, \tau _{2}^{1} =\pi /6 \, , \, \tau _{2}^{2} =\pi /3 \, , \, \tau _{2}^{3} =\pi /2 \, , \, \tau _{2}^{4} =2 \pi /3 \, , \, \tau _{2}^{5} = 5 \pi /6 \, , \, \tau _{2}^{6} = \pi \, , \, \tau _{2}^{7} = 7 \pi / 6 \, , \, \tau _{2}^{8} = 4 \pi /3 \, , \, \tau _{2}^{9} = 3 \pi /2 \, , \, \tau _{2}^{10} = 5\pi /3 \, ,\, \tau _{2}^{11} =11\pi /6$, and so on. In particular for $P=11$ yields $I=0,1,2,3,4,5$ having a circumference placed at the origin, and also five rings having 6, 12, 18, 26 and 32 circumferences distributed along of them as is shown in Fig. 6(a). Therefore, we can write the traditional equation for circumferences, which will form the ideal pattern for designing the null screens leading to
It is important to state that introducing the values for ${\tau }_{I}^{J}$ into Eq. (14) are displayed circumferences starting at $Y$-axis having a clockwise rotation through the subsequent angles ${\tau }_{I}^{J}$, arising from the center to border of the diameter producing Hartmann type null screens. We can define properly the center for each circumference inside the array, which will be useful for evaluate the shape of surface under test as we will explain in the next section. All the circles will be formed as a set of $M$ discrete points for $X_d$ and $Y_d$, which form part of a continuous curve joined with straight lines, yielding a regular polygon with $M$ sides inscribed in a circumference with radius $r_d$. We must keep in mind that by reducing the numbers for $M$, we could accelerate the design of null screens. 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. Finally, the inner area for all circumferences will be transparent and the outer area will be blackened, forming the null screen, which is printed on an acetate foil using a commercial laser printer specified at 2400 dpi. For simplicity, we consider the letter $Q$ to define the total number of circumferences or circular spots, which will be displayed inside of detection plane.
Analogously, to design null screens in such a way that their image formed through refraction yields a uniform array of bright fringes at the detection plane, we start defining a set of points $(X_{d_m},Y_{d_m})$ placed on straight lines. Thus, we defined a uniform interspacing along the $X$-axis, according to $X_{d_m} = (i-1/2)\Delta$, for $i = 0,1,\ldots ,P,$ where $P$ is the number of lines, which will form $Q$ bright fringes, whose separation is given by $\Delta = l_d/(2P-1)$. We have also defined $Y_{d_m}$, formed by a sequence of $m$-points which will form a closed curve for each $Q$-fringe, the height for $Y_{d_m}$ is limited up to the circumference with diameter $l_d$, in other words $Y_{d_m} \in [-([l_d/2]^2-X_{d_m}^2)^{1/2}, ([l_d/2]^2-X_{d_m}^2)^{1/2}]$. To produce the null screen we select two contiguous straight lines and put them together, for instance the set $(X_{d_{m-1}},Y_{d_{m-1}} ) \cup (X_{d_{m}},Y_{d_{m}} )$ form a closed curve in such a way, the inner area will be transparent passing light through it. On the other hand, the set $(X_{d_{m}},Y_{d_{m}} ) \cup (X_{d_{m+1}},Y_{d_{m+1}} )$ will form a closed curve, and the inner area is blackened to obstruct incident light. The process is repeated until the area is completely filled with fringes, providing the ideal pattern for design Ronchi type null screen as is shown in Fig. 6(b).
We have designed null screens considering two ideal patterns to be recorded in a CCD camera as was explained above, assuming a diameter at detection plane given by $l_d=50$ mm in both cases. For designing Hartmann type null screens (HNS), we consider a quasi-angular uniform array, assuming $P=29$, and by substituting these values into Eq. (13), provides $\Delta = 0.877$ mm, and the radius for all circumferences is $r_d=0.439$ mm, obtaining a total of $Q=661$ circles as is shown in Fig. 7(a). Alternatively, for designing Ronchi type null screens (RNS), we assume $P=33$, and substituting into Eq. (13) we get a width $\Delta = 0.769$ mm for each fringe, providing a total of 33 straight lines, and yield 16 black and $Q = 17$ white stripes as is shown in Fig. 7(e). We can see in Figs. 3(a) and 3(b) the behavior for sagittal and tangential caustic surfaces, covering an extensive region along the optical axis, so the detection plane apparently always lies within the caustic surfaces. The coordinates $(X_d, Y_d)$ of the ideal patterns are substituted into Eq. (12) and we can clearly see that the null screen has a dependence of the coordinate $Z_d$, the size of the detection plane $l_d$, and the diameter of the surface under test proving a set of non-linear equations as a function of $(x_i, y_i , {\mathcal {L}}_i)$ coordinates, in such a way that the null screens are unique and works very well if the experimental conditions a priori established are properly fulfilled.
For example, assuming $Z_d = 60$ mm the ideal circles of the HNS become asymmetrical ovals, and we call them drop shaped spots, the HNS is formed by ${\mathcal {Q}}=730$ drop shape spots as is shown in Fig. 7(b), in other words 730 drop spots are overlapped with 661 circular spots at image plane, the central drop shape spots are slightly deformed, because the surface is nearly plane around of the optical axis. Additionally, a RNS become non-uniform closed curved stripes, and we call them curved stripes, the RNS produces ${\mathcal {Q}}=33$ curved stripes as is shown in Fig. 7(f), which should be overlapped with 17 ideal straight fringes. Subsequently, placing the detection plane at $Z_d=150$ mm the HNS is formed by ${\mathcal {Q}}=761$ drop spots as is shown in Fig. 7(c), and the RNS is formed by ${\mathcal {Q}}=41$ curved stripes as is shown in Fig. 7(g). Subsequently, assuming that $Z_d=240$ mm the number of holes for the HNS have substantially increased, which is formed by ${\mathcal {Q}}=905$ drop shaped spots as is shown in Fig. 7(d). Finally, the number of transparent curved stripes forming the RNS has also increased, having now ${\mathcal {Q}}=43$ curved stripes as is shown in Fig. 7(h). It is important to comment that increasing the distance for $Z_d$ the ${\mathcal {Q}}$ number for either drop spots or curved stripes substantially increases and they should coincide with the $Q$ number for either ideal circular spots or bright straight fringes respectively. A graphical visualization for different null screens considering diverse planes of detection are shown in Visualization 2.
Fig. 7. Ideal patterns for designing: (a) HNS. (e) RNS. Design of null screens for $Z_d=60$ mm: (b) HNS. (f) RNS. Design of null screens for $Z_d=150$ mm: (c) HNS. (g) RNS. Design of null screens for $Z_d=240$ mm: (d) HNS. (h) RNS.
6. Qualitative and quantitative tests for a Zernike surface
The images were recorded with a CCD color camera (Thorlabs Model DCU223M), with a sensor of 5.95 mm $\times$ 4.76 mm, attached to a lens with $F=8$ mm focal length, which is used to focus light on CCD. The null screens were printed on an acetate foil and were placed on an acrylic ring base to give them mechanical strength. The light source used is a polarized He-Ne laser ($\lambda = 633$ nm). The diagram of the experimental setup is shown in Fig. 5, where we can also see a polarizer near the laser beam, which has exclusively been used to reduce the amount of irradiance impinging on the CCD, and we have also implemented a collimator lens ($F/\# = 5$) with a large focal distance in order to reduce the hot spots exposed at CCD. The surface under test was mounted on a rotational stage for easy centering along the $X$- and $Y$-directions, furthermore to incline the surface at the right angle. The CCD camera was mounted on a $XYZ$ stage to locate it in such a position that the detection plane can be observed; we have used an opaque screen for recording either bright circular spots or fringes in such a way that the whole surface can be evaluated at once.
After placing the null screens according to Fig. 5, with the detection plane located at $Z_d = 60$ mm, we can see the images recorded on the CCD camera as are shown in Fig. 8(a) and (d), showing an array of bright circular spots and illuminated fringes with a quasi-uniform intensity distribution, respectively. They slightly differ from ideal patterns as are shown in Figs. 7(a) and 7(e), respectively. Additionally, placing the detection plane at $Z_d=150$ mm, the images were recorded as are shown in Fig. 8(b) and 8(e), we can see a non-uniform intensity distribution in many regions of the images due to divergence of the rays passing through the Zernike surface. There are many bright spots on the borders of the image, which are not overlapped at detection plane, and they slightly differs from ideal pattern, the illuminated fringes is less sensitive than the bright circular spots. Finally, considering the detection plane placed at $Z_d=240$ mm, the images are shown in Figs. 8(c) and 8(f), where the borders differ substantially from the ideal pattern because the bright circular spots are not in coincidence as we expected. Furthermore, there exist bright fringes that resemble four focal regions. We can also see that both images have non-uniform intensity distribution, and make it difficult to obtain the centroids of the images.
Fig. 8. Images recorded experimentally placing the detection plane at $Z_d=60$ mm for: (a) HNS, and (d) RNS. Placing $Z_d=150$ mm for: (b) HNS, and (e) RNS. Placing $Z_d=240$ mm for: (c) HNS, and (f) RNS, showing that the intensity distribution becomes irregular.
The test is very sensitive to alignment and positioning. For example, in Figs. 9(a) and 9(e) the images were recorded considering an incident plane wavefront refracted through the Zernike surface without null screens in such a way that the detection plane was placed very far from the surface under test, $Z_d=330$ mm and $Z_d=2360$ mm, respectively. We can see effects of diffraction and the formation of caustic surfaces in Fig. 9(e). In particular, the Fig. 9(a) resembles a zoom of the Fig. 3(e) as we expected. Figure 9(b) was recorded at $Z_d=50$ mm with a null screen rotated around the Y-axis (we called tilt misalignment) an angle $\theta _y = 17^{\circ }$, while Fig. 9(f) the null screen was rotated $\theta _y = 38^{\circ }$ with the detection plane placed at $Z_d = 150$ mm. The images with tilt misalignment produce an elliptical boundary registered at CCD as is shown in Figs. 9(b) and 9(f). On the other hand, in Fig. 9(c) the null screen was rotated around the $Z$-axis (we called rotational misalignment) an angle $\theta _{z}= 45^{\circ }$ and recorded considering $Z_d = 150$ mm. Subsequently, in Fig. 9(g) the rotation angle is $\theta _{z}= 36^{\circ }$ placing the detection plane at $Z_d = 240$ mm. The null screen in Fig. 9(d) was designed to be placed at $Z_d = 240$ mm, but it was recorded at $Z_{d} = 50$ mm. Additionally, in Fig. 9(h) also shows a rotational misalignment, in this case $\theta _{z}= 23^{\circ }$, considering that the detection plane was placed at $Z_d = 60$ mm. It is important to comment that we can see a quasi-uniform intensity distribution on the images.
Fig. 9. Surface illuminated without null screen for (a) and (e) placed at $Z_d=330$ mm, and $Z_d=2360$ mm, respectively. (b) and (f) present tilt misalignment. (c), (g), (d) and (h) present rotational misalignment having different $Z_d$ distances.
We have considered Fig. 8(a) to retrieve the centroids as are shown in Fig. 10(a) by using an image-processing program, which considers for all pixels their intensity value recorded by the CCD. All the centroids were corrected for the lens distortion introduced by the CCD camera ($E=-2.6509\times 10^{-5}$ mm $^{-2}$) as is shown in Fig. 10(b). The next step is calculating the approximated normal vectors to the surface under test, according to the formula
Fig. 10. (a) Centroids in the CCD sensor plane. (b) Centroids and their ideal position. (c) Integration paths to evaluate the surface under test.
By using the calculated normal vectors the shape of the surface is obtained through the integral $z = z_0 - \int _{P_0}^{P} ( [{n_x}/{n_z}] dx + [{n_y}/{n_z}] dy )$, or alternatively, the simplest method for the numerical integration is the trapezoidal rule leading to
Fig. 11. (a) Discrete points and the retrieved surface. (b) Difference between ideal versus fitted surface. (c) Elevation maps for the differences between ideal versus fitted surface.
Table 1. Parameters Obtained From the Nonlinear Least Square Fitting of the Experimental Data
From Table 1, the fitted parameter $F_1$ differs by approximately 0.021 or about $0.015\%$ of the ideal value, for the parameter $F_2$, it differs by 7.848 or about $1.168\%$. For the parameter $F_3$, it differs by 6.771 or $0.671\%$, finally, for the parameter $F_4$ it differs by 8.754 or about $1.824\%$. We can clearly see in Fig. 11(a), the retrieved surface for the best fitting and the set of discrete data numerically obtained by the integration process for the surface under test. In addition, the differences in sagitta between the ideal Zernike surface and the best fitting freeform surface is shown in Fig. 11(b), showing greatest departures at the borders of the surface. In this case the P-V differences in sagitta between the evaluated points and the best fitting is $\delta z_{PV} = 0.03$ mm, and the RMS difference in sagitta value is $\delta z_\textrm {RMS} = 6.3 \mu$m. Then the null screen method allows measurement of the shape of the surface with medium precision [23]. Here, departures from the perfect shape have been clearly observed. We present a color contour map of the differences between ideal and fitted surface as is shown in Fig. 11(c).
It is important to state the following: there are slight non-uniformities of intensity displayed on the opaque screen and 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 laser beam and passing through the collimator lens producing a quasi-monochromatic and quasi-plane wavefront propagating along the experimental setup, which could slightly modify the images recorded. Additionally, we have printed the null screens on acetate foil, assuming that the acetate foil is a quasi-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 for the null screens on the foil surface, these imperfections can considerably alter the images recorded on the sensor due to errors of printing. The surface departures from the best surface fit are of the order of one $\mu$m 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 screens are less than $0.5$ mm, according to [23]. Furthermore, we are assuming that the first face of the Zernike surface completely is plane. Certainly, possible slight misalignment in the experimental setup could also modify the images recorded.
7. Conclusions
In this manuscript, we have implemented an exact ray trace through a plano-freeform Zernike surface, that allows us to provide both sagittal and tangential caustic surfaces for a plane wavefront propagating along the optical axis. We provide an exact formula to represent their wavefronts propagated arbitrary distances outside of the Zernike surface. We believe that the method for obtaining the caustic surfaces reported here are straightforward, giving formulas to represent sagittal and tangential caustic surfaces. Additionally, we have properly implemented a quantitative test to evaluate the shape of the Zernike surface, based on the method of null screens, placing the detection plane inside the caustic surfaces. We showed that, with this method, it is possible to evaluate, in a simple manner, the shape of plane-curved freeform surfaces with large deformations, with PV values greater than $0.43$ mm, and values greater than 6.3 $\mu$m in $\delta _\textrm {RMS}$. Another advantage of this work is that with this alternative method we can design arbitrary null-screens and their image that will be observed on the detection plane is displayed in real time. We recovered with a simple process the Zernike coefficients making a polynomial fit and reducing the errors added with these issues. The work presented here opens the door to test freeform surfaces using null screens with the detection plane inside the caustic surfaces.
Funding
Consejo Nacional de Ciencia y Tecnología (A1-S-44220); Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IN116420).
Acknowledgments
The authors are indebted to Richard N. Claytor and D. Nelson E. Claytor from Fresnel Technologies Inc. for providing us the Zernike surface to implement the test of this manuscript.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
No data were generated or analyzed in the present research.
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