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Abstract
In this work, we present the fabrication of two 3D printed plano-freeform prototypes, designed in such a way that, when assembled, an Alvarez lens is formed. The freeform surface of each element was mathematically described using Zernike polynomials and verified by implementing an off-axis null-screen test. Additionally, a characterization by refraction of the assembled lens was performed. Experimental images show the suitability of additive manufacturing engineering for prototyping freeform optics by providing a practical demonstration of the Alvarez lens concept.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Freeform optics has emerged as a groundbreaking technology that has revolutionized imaging and illumination optics. Due to this, design requirements have become increasingly complex, leading to a growing trend toward using freeform surfaces that fulfill demands. However, the complexity of these freeform surfaces has also posed new challenges in the design, fabrication, and testing since they are very difficult or almost impossible to fabricate and test using traditional methods, opening an excellent opportunity for expanding research and development in these lines of study. Fabrication of complicated optical surfaces has become possible using ultraprecision diamond turning machines [1,2]. By using a Computer Numerically Controlled (CNC) process developed by Optimax Systems, Inc (Ontario, NY, USA), freeform optics elements are generated, polished, and evaluated [3]. Balancing gravitational force, hydrostatic pressure, and surface tension in a cylindrical bounding frame, freeform components are fabricated by fluidic shaping [4]. Stereolithographic fabrication has been studied for manufacturing terahertz optics with arbitrary shapes [5]. Furthermore, additive technology has been used to fabricate biconvex aspherical prototypes that are used in an indirect ophthalmoscope [6,7]. This same technology has also been applied to produce Alvarez lenses [8].
On the other hand, to prove and guarantee the method’s functionality, no matter which fabrication process is, a measurement of the final piece must be realized to ensure the prototype meets the design parameters. In Ref. [1], a Scanning Electron Microscope (SEM) and an Atomic Force Microscope (AFM) were used for the prototypes’ testing. Weck et al., in [2], used an optical system mounted on the rotating milling tool for measuring its geometry. The metrology of the freeform surface fabricated by Blalock et al., in [3], was performed with a Coordinate Measurement Machine (CMM), Interferometry, and Computer Generated Holograms (CGH). The topography of the freeform surface fabricated by Elgarisi et al., in [4], was measured using a Digital Holographic Microscope (DHM). Graves et al. introduced Infinite Deflectometry, a technique employed to evaluate a 6 mm Alvarez lens with $17 \mu$m of horizontal coma and $-17 \mu$m of trefoil that represents a highly freeform surface which presents a unique metrology problem [9]. In [10], Kim describes improvements to deflectometry systems designed for the metrology of challenging optic shapes. Aguirre et al., in [11], used the null screen method for the freeform surface evaluation.
As can be seen, several fabrication techniques and evaluation methods have been studied, developed, and implemented for freeform surfaces. Therefore, the ideal method for fabrication and metrology must be chosen considering the application and design specifications of the surface in question.
In this work, we focus on fabricating freeform optics based on the concepts of Alvarez and Humphrey lenses, which have gained significant attention in recent years for their applications in both imaging and non-imaging optics [12–19]. Some other advanced and modern applications that use Alvarez’s concept is virtual reality systems [20]. In recent years, there have been developments in printed three-dimensional freeform gradient index optics [21,22].
The Alvarez lens concept, originally developed for ophthalmic applications, involves a two-element lens system. Both elements are displaced an equal distance in opposite directions along the axis of lateral symmetry to introduce variable optical power [23–25]. The conditions and examples of laterally and rotationally translating variable power lenses can be found in Refs. [26,27].
We present the prescription, fabrication, and testing of two 3D printed plano-freeform prototypes, that when assembled, form an Alvarez lens. The printed prototypes are complementary elements that, when one element is displaced relative to the other one, the magnitude of the optical effect changes. Depending on the direction of the movement, second-order aberrations, including defocus and astigmatism, are introduced. Additionally, we propose a metrology method for the freeform surface shape of the prototypes, implementing an off-axis null screen test by reflection.
The work is organized as follows: In section 2, the prescription used to generate the 3D model of the plano-freeform element is presented. In section 3, we describe the experimental setup and considerations of the 3D printer, as well as the post-processing procedures. Section 4 presents the metrology of the freeform surface shape of both prototypes using a null screen test by reflection, as well as the characterization of the optical power as a function of lateral displacement. In section 5, images through both prototypes are presented. Finally, conclusions and some final comments are presented in section 6.
2. Mathematical model
Before starting the fabrication process of the freeform elements, the three dimensional model of the prototype’s shape is needed. In this section, we present the mathematical model for the freeform surface of the 3D printed elements, and the parameters that characterize the final pieces.
As it was mentioned before, a two-element lens is proposed to achieve the effect of change in optical power by laterally translating the elements by an equal and opposite amount. The mathematical description of the well known Alvarez-Humphrey lenses was used to describe the freeform surface of both elements. By rewriting the lens equation described in the patent [24] in terms of Zernike polynomials, the final expression becomes a sum of an equal amount $a=a_6=a_7$ of vertical trefoil and vertical coma, respectively.
by substituting the ANSI Zernike polynomials and simplifying terms, the mathematical expression in cartesian coordinates is given by Eq. (2) where $a$ is an arbitrary coefficient, $X$ and $Y$ are normalized coordinates, and $F(Y)=-3\sqrt {8}aY$. The representation of the Zernike polynomials employed and the elevation map of the final surface are shown in Fig. 1.
Fig. 1. Zernike polynomials and freeform surface colormap. (a) Trefoil ($a_6=0.43$), (b) coma ($a_7=0.43$), and (c) colormap of the final prototype in normalized coordinates.
The other element is complementary to the first one $t_2(X,Y)=-t_1(X,Y)$. Therefore, the final contribution of both surfaces is proportional to
Depending on the amount and direction of lateral shift ($\Delta X$ or $\Delta Y$), defocus or astigmatism can be added. It is important to mention that for this case, according to the coordinate system shown in Fig. 1, the change in optical power is achieved by lateral displacement in $Y$ direction.
To derive the expression describing the change in optical power as a function of lateral displacement, we begin by setting the displacement in the $X$ direction to zero, $\Delta X = 0$. After substituting the Zernike terms and simplifying the equation, the resulting expression accounts for the combined elements, that can be approximated as a parabolic term that is equivalent to a spherical wavefront. The approximation in non-normalized coordinates takes the form $t(x, y) \approx ((24 \sqrt {2} a \Delta y) (x^2 + y^2))/r_N^3$, where $r_N$ represents the normalization radius, $a$ is the Zernike coefficient, and $\Delta y$ denotes the lateral shift. Assuming elements are in air and applying the first-order paraxial approximation, the optical power in diopters can be determined using the following expression
where the lateral shift $\Delta y$, the normalization radius $r_N$ and the coefficient $a$ are in mm, and $f$ is the focal length in meters. From Eq. (4) it can be noticed that the range of achievable focal lengths depends on the index of refraction of the material $n$ and the coefficient $a$. The greater the coefficient, the higher optical power can be obtained through lateral displacement of one element relative to the other.Once the mathematical description of the freeform surface is defined, the stereolithography ".STL" file of the plano-freeform element is generated by using Ansys Zemax OpticStudio (Canonsburg, PA, USA) and AutoCAD (Autodesk Inc, San Francisco, CA, USA). It’s worth noting that Zemax OpticStudio was only utilized for modeling the final prototype and generating the standard tessellation language file (.STL) for 3D printing. To represent the freeform surface described by Eq. (1), a Zernike Standard Sag was employed as the surface type. In alignment with the software’s Zernike convention [28], the terms utilized were Z7 (representing coma) and Z9 (representing trefoil). As a result, the coefficients used in the software were $a=a_7=a_9$ and the normalization radius was $r_N=20$ mm. The two elements that assemble the lens are printed with the same file because when the experimental setup is mounted, a rotation of 180 degrees in $Z$ is enough to achieve $t_2(X,Y)=-t_1(X,Y)$. As was mentioned above, the other surface of both elements is a plane.
To show the feasibility of fabricating and testing freeform surfaces, and that the fabricated prototypes performed according to the Alvarez lens concept, we select a parameter $a=0.43$, corresponding to a peak-to-valley sagitta of $3.73$ mm. The refractive index used, previously characterized for the 3D printer’s clear resin [29], is $n=1.505$ for a wavelenght $\lambda =632.8$ nm, with a maximum lateral displacement of the elements set at $\Delta y=\pm 5$ mm. These parameters were selected based on laboratory conditions. The resulting optical power range achieved under these specifications extends from $-9.2$ D to $9.2$ D, corresponding to a focal length range from $-\infty$ to -108.5 mm and from 108.5 mm to $\infty$. Detailed specifications of the elements are listed in Table 1.
Table 1. Element prescription
In the following section, the printer parameters and the post-processing method to fabricate the 3D printed prototypes are explained.
3. Fabrication process
For the fabrication of the 3D printed plano-freeform elements, we used the Formlabs (Somerville, MA, USA) Form 3 stereolithographic printer with Clear Resin, both previously characterized [29]. To achieve a superior finish on the 3D printed prototypes, we examined various variables during the manufacturing process. To prevent obstruction of the pupil of the prototypes, we strategically placed the printing supports along the edges of the pieces. Additionally, we discovered that layer errors are minimized when the printer’s bed is angled in the $Y$ direction with respect to plane $XY$. The final prototypes were fabricated using the parameters listed in Table 2.
Table 2. Printer parameters
After finishing the printing process, a bath with isopropyl alcohol (IPA) is required to clean the resin excess; the same company’s Form Wash kit was used. The curing process was carried out with the Form Cure equipment for 15 minutes at a temperature of 60 degrees Celsius. To obtain the required transparency of the final prototypes, two post-processing techniques were made, one for the plane surface and the other one for the freeform surface. The freeform surface post-processing consists of grinding and polishing using micro-graded wet/dry polishing paper, (30,15, 9, 3, and 2 $\mu$m), and aluminum oxide micro-graded polishing paper of 1$\mu$m, following the protocol described in Ref. [7]. When the freeform surface polishing is finished, the plane surface is cured by placing it and pressing it against a glass surface of flatness $\lambda /5$ as is described in Ref. [30].
In Fig. 2(a) the prototype extracted directly from the printer after finishing the printing is shown. An intermediate stage of the post-processing can be observed in Fig. 2(b). The final finishing of one of two elements is shown in Fig. 2(c).
The post-processing technique was applied to both elements, yielding favorable outcomes for the two final pieces. Through a qualitative analysis, when we look at the image of a perfect grid through the first element, the lines appear to curve upward, as shown in Fig. 3(a). In contrast, the image obtained from the second element shows the contrary effect, with the lines curving downward, as seen in Fig. 3(b). But, when these two elements are assembled, this effect is counteracted, resulting in straight lines, as demonstrated in Fig. 3(c).
Fig. 3. Final 3D printed and polished prototypes. (a) Prototype 1, (b) prototype 2, and (c) prototypes assembled.
While the qualitative evaluation of the 3D-printed polished prototypes yielded promising preliminary results using additive manufacturing technology, it is essential to verify their surface shape and characterize their performance. The next section provides a detailed analysis of the evaluation process.
4. Metrology of the 3D printed prototypes
Considering the complexity of the shape of the freeform surface, the metrology of the prototypes was realized by implementing an off-axis null screen test by reflection, addressing all the challenges associated with both the evaluation method and the freeform surface complexity. The key advantage of employing the null-screen test lies in its simplicity; it can be readily implemented without the need for null lenses or compensators. Moreover, it eliminates the risk of surface damage due to direct contact, making it a practical and effective solution for our evaluation needs. Additionally, a characterization through refraction of the assembled lens was performed, determining the variation in optical power as a function of the lateral displacement between the two elements.
4.1 Off-axis null screen test by reflection
The off-axis null screen method is a non-contact metrology test that consists of designing a target that after reflection over an ideal surface, leads to a uniform pre-designed pattern [31]. Using image processing, the experimental centroids of the image captured by the detector are found, and the approximated local normal vectors at the surface are computed; finally, the surface shape is recovered by numerical integration. It has been proved that for rotationally symmetric surfaces if the system is perfectly aligned, the differences between the experimental centroids and the ideal pattern indicate surface deformations.
However, one caveat of this test is that, although the system is perfectly aligned and the element under test does not have any deformation, the obtained images can be misleading. In particular, for testing freeform surfaces it has been found that depending on the geometry of the setup, although the null-screen was designed to obtain an uniform image, some blur could be present [32–34].
The parameters selected for fabricating the prototype were chosen to show the feasibility of manufacturing and testing highly freeform surfaces. Specifically, the parameter $a$ employed to define the freeform surface corresponds to a peak-to-valley in sagitta of 3.73 mm. This type of surface allows us to demonstrate the importance of the element’s positions for a successful evaluation using the null-screen technique. As usual, the elements involved in the test are a CMOS camera, the surface under test and the null-screen. In this work, a plane null screen was used, giving the advantage of using an electronic device as the screen that allowed designing and displaying different null screens without needing to make any mechanical change in the calibrated experimental setup.
The position of each element used in the experimental setup was calculated through computer simulations to obtain the best experimental image. The simulated images were obtained by following the approach outlined in Ref. [11]. The test geometry is intentionally selected to ensure a clear pattern in the resulting image, in this way the centroids can be easily found. Figure 4 shows the designed null-screen and its correspondent simulated image at different angles of the surface with respect to the test’s elements.
Fig. 4. Null-screen (above), reference surface (middle) and simulated image (below) for null screen testing of an Alvarez plate $a=0.43$ at $\theta _x=55^{\circ }$ for (a) surface under test at $\theta _z=0^{\circ }$, (b) surface under test at $\theta _z=45^{\circ }$, (c) surface under test at $\theta _z=90^{\circ }$ and (d) surface under test at $\theta _z=180^{\circ }$.
It is very important to clarify that all the calculations were computed assuming that the origin of the reference system is set at the center of the surface, and that the $XY$ plane is defined to be parallel to the plane face of the prototype. In this off-axis test example, to easily instrument the test, the $XY$ plane of the reference system is defined at $\theta _x=55^{\circ }$ with respect the optical table by using a 3D printed wedge previously characterized; this allows to have the $Z$-axis of the reference system at $145^{\circ }$ from the optical table, as it is shown in Fig. 5(a). The surface under test was rotated around the $Z$-axis by an angle $\theta _z=45^{\circ }$ to obtain a clear pattern in the image avoiding ambiguities. To achieve the rotation around the $Z$-axis, the surface under test was mounted on a rotation stage, in this way, the reference system undergoes a simultaneous rotation along with the prototype, without affecting the previously calibrated elements.
The camera’s position is defined by a vector parallel to its optical axis, extending from the pinhole to the system’s origin, the sensor was separated from the surface’s center in vertical direction a distance $d_{z}=$103 mm, this is a key feature because this distance has to stay fixed due to camera calibration for distortion. Meanwhile, the null-screen’s position is defined by a vector perpendicular to the screen and separated from the surface’s center in horizontal direction a distance $d_y=13$ mm.
The CMOS camera and the cell phone used to display the null screens were mounted on translation stages. This allows aligning perpendicularly the screen and parallelly the camera sensor, both to the optical table. To ensure that the results obtained are attributed only to the surface shape and not affected by alignment issues, the system was previously characterized, and the camera was calibrated for distortion following the procedures outlined in Ref [35].
The elements used in the experimental setup were: a CMOS camera with a lens with focal length of $f=8$ mm and a sensor’s size of 4912$\times$3684 pixels (6.1$\times$4.6 mm), and a pixel size of 1.25 $\mu$m. To display the calculated null-screen, a Nokia 3 cellphone with an LCD screen of 1280$\times$720 pixels (110.74$\times$62.23mm), with a pixel size of 86.5 $\mu$m. The experimental setup is shown in Fig. 5(b).
The null-screen was designed by following the steps outlined in Ref. [36]. In summary, once the ideal pattern and the positions of the elements involved were set, an inverse raytracing was performed. The incident rays were described by equations of straight lines defined by discrete points over the sensor and the pinhole position $P_{ph}$. Then, intersections of the incident rays with the reference surface were found, and the local normal vectors were evaluated at these points $P_{int}=(x_{int},y_{int},z_{int})$ (Eq. (5)).
To find the reflected rays $\boldsymbol {\hat {R}}$, the Vectorial Law of Reflection (Eq. (6)) was applied
Because the two elements that constitute the Alvarez lens should be identical, the same null screen and experimental setup can be used to test the freeform shape of both prototypes. The back surface of the prototype was coated with paint to prevent spurious reflections. Images of the null-screen after reflection on each surface under test were captured and analyzed to determine the surface shape of the each prototype.
After performing image processing of the experimental images shown in Fig. 6, the experimental centroids of each spot were obtained. Reference [37] discusses the impact of errors in the determination of centroid positions. By finding the experimental incident vectors $\boldsymbol {\hat {I}_{exp}}$, with the experimental centroids and the pinhole position, the intersection with the ideal surface was obtained.
The new intersection points and their corresponding object point in the null-screen define the equations of straight lines, which will be the experimental reflected rays $\boldsymbol {\hat {R}_{exp}}$.
The experimental normal vectors are approximated by applying Eq. (7).
Once we collected the experimental data, the trapezoidal rule Eq. (8) was applied by using different integration paths, recovering discrete points that describe the surface shape of each tested prototype
The cloud of discrete points, labeled as $z_m$, which were obtained through zonal integration, were fitted to a 15 ANSI Zernike polynomials using a normalization radius of $r_N=20$ mm. The coefficients obtained for each prototype are shown in Fig. 7.
Results show that the coefficients of coma ($a_7$) and trefoil ($a_6$) stand out as the dominant Zernike terms, with their values closely aligned to the nominal specifications. While there exist minor contributions from other coefficients, these are notably smaller in magnitude when compared to the dominant coefficients.
The color map shown in Fig. 8(a) represents the ideal surface shape of the fabricated element, while Figs. 8(b) and 8(c) display the surface maps of the recovered freeform surfaces for the 3D printed prototypes. The discrete points $z_m$ are represented by the black dots.
Fig. 8. Freeform surface. (a) Ideal surface shape, (b) recovered prototype 1, and (c) recovered prototype 2.
To analyze the deviations between the 3D printed prototypes and the ideal surface shape, the variations between the ideal and experimental sagittas were computed and they are show in the plot depicted in Fig. 9. For prototype 1, the RMS found was 0.1442 mm, meanwhile the PV value was 0.8512 mm. For prototype 2, the RMS was 0.1334 mm and the PV value was 0.7342 mm. While these values may appear high, it is important to recall that manual post-processing was performed, and the observed degradation at the edges is a common occurrence, as depicted in Fig. 9.
Furthermore, the order of magnitude of both differences is comparable, offering valuable insights into the fabrication process, which could potentially be mitigated in future trials. An analysis comparing the ideal and experimental coefficients was also performed. A chart showing these differences for each prototype is shown in Fig. 10.
Fig. 10. Differences between ideal and experimental coefficients for prototype 1 (blue), and prototype 2 (orange).
It is worth mentioning that, for the purpose of validating the null screen test results, we measured prototype 1 using an alternative technique, a Zeiss Coordinate Measuring Machine (CMM) Model MC850, Serial number 71762. The discrete data obtained were fitted to the same ANSI polynomial, and the resulting coefficients as well as the differences between both methods are detailed in Fig. 11.
Fig. 11. (a) Zernike coefficients for prototype 1 measured with null-screen test (blue) and measured with CMM (purple). (b) Differences between both methods.
Examining the differences between the coefficients shown in Fig. 11, it can be seen that the differences for the coefficients responsible for shaping the overall element $a_6$ and $a_7$ are $\Delta a_6 = \Delta a_7=0.1$. Nevertheless, the most significant differences corresponds to coefficient $a_5$, vertical astigmatism. This discrepancy could be attributed to slight differences in data alignment and potential compensation between the Zernike coefficients.
Note that, the results of both methods closely coincide, approaching the nominal description. Consequently, the calibrated null-screen testing system described earlier was established and employed to evaluate the final 3D printed Alvarez lens prototypes.
4.2 Power vs lateral displacement analysis
After testing the shape of the freeform surface, a characterization of change in optical power as a function of lateral displacement of the Alvarez lens was realized. To test the performance of the lens, an HeNe laser of $\lambda =632.8$ nm was collimated; each element under test was mounted in a rotation stage with clear aperture of 25.4 mm in diameter and placed in the collimated beam. It’s worth noting that the lenses are highly sensitive to alignment; therefore, precision translation stages with motions in $x$, $y$ and $z$ axes were employed to improve alignment.
Figure 12 shows the experimental setup. To perform lateral shifts in the $Y$ direction, each element of the Alvarez lens was mounted on a linear translation stage. It is important to highlight that, in order not to move the optical axis of the assembled lens, each element has to be moved the same amount $\Delta y$ in opposite directions, as specified by Eq. (3). Once the system was aligned, the paraxial focal length $f(\Delta y)$ was found for each controlled lateral shift. For measuring the range of negative focal lengths, an auxiliary spherical lens of $f_{aux}=100$ mm was used.
By plotting the optical power $\phi$ in diopters as a function of lateral shift, assuming that $\phi =1/f(\Delta y)$, we can find that the optical power of the Alvarez lens as a function of lateral shift follows a linear equation as it shown in Fig. 13, this behavior is consistent with Eq. (4).
The experimental data was fitted to a linear model to obtain the slope $m$. Once the slope was found, the following equation was used to find the experimental coefficient of both elements
where $m$ is the slope of the lineal model, $r_{N}=20$ mm is the normalization radio or semidiameter, and $n$ is the refractive index. Substituting the values in Eq. (9), a coefficient of $a=0.44$ mm was recovered. It is important to mention, that an approximation of both elements’ coefficients applying this method was found. Although the elements are not exactly identical, the overall effect and behaviour of both elements are considered as a whole. As well as the fact that since these lenses are typically used for ophthalmic applications, it is expected that the refractive index does not significantly impact their performance. In the next section, we show the results obtained with the final prototypes.5. Experimental images
As it was described in section 2, the amount and direction of lateral shift applied to the prototypes can introduce defocus or astigmatism. In this section, experimental images captured using the 3D printed prototypes are presented to demonstrate these effects.
The initial step involves aligning the prototypes and precisely defining the reference system, this is because the introduced effects vary depending on the direction of lateral movements. To begin, the alignment of the first element is performed, Fig. 14(a) shows the output beam from the first element. Because the apex of the triangle corresponds to the convex part of the prototype, the element is oriented as it is shown in Fig. 1(c). Figure 14(b) shows the output beam from the second element, oriented $180^{\circ }$ from the first one. When the two elements are assembled, a collimated beam is observed as it is shown in 14(c). To perform this task, each element was mounted in a rotation stage, in this way the angle of each element is controlled and measured to achieve the different effects.
Fig. 14. (a) Alignment of the first element, (b) alignment of the second element, (c) collimated beam through both elements, (d) convergent beam obtained by lateral displacement $\Delta y>0$ in $Y$ direction, (e) diverging beam obtained by lateral displacement $\Delta y<0$ in $Y$ direction, (f) $-45^{\circ }$ astigmatism obtained by lateral displacement $\Delta x>0$ in $X$ direction, and (g) $45^{\circ }$ astigmatism and defocus obtained by lateral displacement $\Delta x<0$ in $X$ direction.
Once the prototypes are oriented and aligned, controlled equal lateral movements of both elements $\Delta y$ are performed following Eq. (3). When the elements are moved $\Delta y>0$ in $Y$ direction, the lens works as a positive lens, and a convergent beam is obtained as it is shown in Fig. 14(d); depending on the amount of $\Delta y$ the focal length of the lens changes. On the other hand, if both elements are moved $\Delta y<0$, the lens works as a negative lens, and a divergent beam is obtained, as it is shown in Fig. 14(e). To achieve astigmatism, the movement of the elements has to be performed in $X$ direction. If $\Delta x> 0$, astigmatism at $-45^{\circ }$ is obtained, as it is shown in Fig. 14(f). If $\Delta x<0$, astigmatism at $45^{\circ }$ is obtained, as it is shown in Fig. 14(e).
Figure 15 illustrates the image quality of the 3D printed prototypes through a series of images. Each image was captured by laterally shifting one element relative to the other one in $\pm Y$ direction by steps of one millimeter $\Delta y= 1$mm, resulting in a variation of focal length. It is worth noting that the object distance remained constant for each captured image; consequently, the camera’s focus had to be readjusted for each image, leading to variations in image magnification. The object used to test the image quality of the prototypes was a negative 1951 USAF Resolution Test Target from Thorlabs (Newton, NJ, USA).
Fig. 15. Series of images with varying magnification due to a change in focal length, using the negative 1951 USAF Resolution Test Target from Thorlabs. (a) $\Delta y=-2$mm, (b) $\Delta y=-1$mm, (c) $\Delta y=0$mm, (d) $\Delta y=1$mm and (e) $\Delta y=2$mm.
To qualitatively observe the effect of astigmatism at $-45^{\circ }$ and $45^{\circ }$, images of a grid were captured by moving the elements in the $X$ direction. Figure 16 illustrates the noticeable change in the direction of the lines.
Fig. 16. (a) Image obtained introducing astigmatism at $-45^{\circ }$, (b) elements $\Delta x=0$ and $\Delta y=0$, and (c) image obtained introducing astigmatism at $45^{\circ }$.
To demonstrate the introduction of astigmatism in other directions, the elements were aligned at $45^{\circ }$, as shown in Fig. 17(a) and 17(b). When $\Delta x=0$ and $\Delta y=0$, a collimated beam is obtained, as shown in Fig. 17(c). When the elements are in this position and a movement $\Delta x$ is applied in the $X$ direction, the two-element lens introduces defocus and either vertical or horizontal astigmatism, as seen in Fig. 17(d) and 17(e), respectively.
Fig. 17. (a) Alignment of the first element, (b) alignment of the second element, (c) collimated beam through both elements, (d) horizontal astigmatism and defocus obtained by lateral displacement in $-X$ direction, and (e) vertical astigmatism obtained by lateral displacement in $X$ direction.
This effect can be observed in the images of a grid shown in Fig. 18. It should be noticed that in this case, the lines elongate in the horizontal and vertical directions, respectively.
Fig. 18. (a) Image obtained introducing horizontal astigmatism, (b) elements $\Delta x=0$ and $\Delta y=0$, and (c) image obtained introducing vertical astigmatism.
It is important to emphasize that proper alignment and the direction of lateral movements are essential for achieving the expected results. Finally, the images show in this section demonstrate successfully the functionality of the 3D printed prototypes and the image quality achieved through both elements.
6. Conclusions
In this work, a commercial lithographic 3D printer for the fabrication of Alvarez lenses prototypes was used. The freeform shape of the fabricated prototypes was described by Zernike polynomials, and their surface shape was validated through the implementation of an off-axis null screen test. The findings reveal that the root mean square (rms) discrepancies in the design parameters and the 3D printed element were 0.1442 mm, meanwhile the PV value was 0.8512 mm for prototype 1; and RMS was 0.1334 mm and the PV value was 0.7342 mm for prototype 2. We presume that these values result from manual post-processing, yet they provide valuable insights into the fabrication process. However, when analyzing the two-element lens as a whole, it effectively demonstrates the overall concept of an Alvarez lens. The transparency achieved in the 3D printed prototypes contributes to a high level of image quality, providing evidence that 3D printing is a viable method for manufacturing optical elements with intricate fabrication requirements.
Funding
Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IT100321, IT103823).
Acknowledgments
The primary author extends gratitude to Prof. Jim Schwiegerling for his valuable insights into Alvarez lenses. DGU acknowledges the support of a fellowship from the CONAHCyT- Postdoctoral Fellowship Program at ICAT-UNITA, UNAM.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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