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The newly formulated theory of aberration fields of freeform surfaces describes the aberrations that freeform Zernike polynomial surfaces can correct within folded powered optical systems. This theory has guided the design of an OLED-based reflective freeform electronic viewfinder covering a 25° full field-of-view with a 12 mm eyebox, which is reported together with a detailed methodology that begins with developing an unobscured starting point and ends with an optimized freeform design, analyzed both in display and visual spaces. In addition, tolerancing of the system points to the potential low sensitivity of these systems to manufacturing tilt (10 arcmin), decenter and despace (100 µm), and figure errors (λ/2 @ 0.632 µm).
© 2015 Optical Society of America
1. Introduction
In 1972, Polaroid introduced the SX-70 single lens reflex Land camera, representing the first commercial product using rotationally nonsymmetric optical components, referred to then as nonrotational aspheric surfaces [1]. The surfaces were described by low-order XY-polynomials, a 2D generalization of an aspheric surface description first used by Ernst Abbe [2]. A unique aspect of this camera was its ability to fold flat and, therefore, it required a viewfinder with a reflector set at an odd angle and two rotationally nonsymmetric lenses to relay the light, as indicated in Fig. 1. It was for this camera that the transformative benefits of freeform surfaces, defined here as surfaces without rotational or translational symmetry where the asymmetry extends beyond anamorphic surfaces (a cylinder being a special case), were first realized. Within the last decade, computer controlled machining processes have advanced to a point where fabricating freeform surfaces is no longer prohibitive [3], thereby generating strong interest in the many fields related to freeform surfaces. Active research areas include optical system design with freeform surfaces guided by the aberration fields of freeform surfaces [4–6], freeform surface fabrication [7], freeform surface metrology [3, 8–10], and freeform optics assembly and testing [11].
Fig. 1 Cross-sectional view of the Polaroid SX-70 camera. Released in 1972, this camera represented the first commercial product leveraging freeform optical surfaces. The freeform lenses were low-order XY-polynomials and are circled above. (Image adapted from [1])
For decades, optical system design has been done in accordance with design principles based on rotational symmetry, which greatly simplifies the theory and process. This is more than adequate for the majority of imaging applications, however, for applications that place strict guidelines on package size and/or geometry, it is often necessary for the optical system to operate in an off-axis configuration, breaking the rotational symmetry, and, thus, changing the required design principles and aberration theory.
A major breakthrough was recently made when Fuerschbach et al. derived a formalism for the aberration theory of systems with freeform surfaces, specifically, surfaces described by Zernike polynomials [12]. The theory builds upon work done by Thompson [13], which extended the wave aberration theory for rotationally symmetric systems to systems that are rotationally nonsymmetric, but consist of rotationally symmetric surfaces. With this new theory, a designer is now equipped to undertake the challenging task of optical system design with freeform surfaces.
A previous two-mirror freeform design work of a near-eye display was reported in [4] along with its final design performance for manufacturing. In this paper, we will focus on a two-mirror freeform design for a viewfinder with stringent requirements for eyebox size and telecentricity. Importantly, and novel to this paper, we offer insight into how the design progresses from a first-order starting point we conceived to a final unobscured off-axis design utilizing freeform surfaces. Specifically, the design process and the final performance of an unobscured, reflective, Zernike polynomial-based electronic viewfinder that encompasses a full diagonal field-of-view (FOV) of 25° with an eyebox diameter of 12 mm is reported. Building on the recent reporting of the design of a viewfinder [14], in this paper we discuss pupil specifications in Section 2, provide an overview of freeform aberration theory in Section 3.1, elaborate on the development of the starting point selection in Section 3.2, detail the design steps from the starting design to the final design in Section 3.3, provide a visual system evaluation in Section 4, and finally report on how the system sustains performance towards manufacturing through a thorough tolerance analysis in Section 5, before concluding the paper in Section 6. The design process, shown to be guided by freeform aberration theory and illustrated using the analytical power of CODE V®’s Full Field Displays (FFDs), demystifies the design process of freeform systems.
2. Electronic viewfinder design considerations
A viewfinder is an optical system, typically found on a camera, which allows the photographer to see the scene as it would be photographed. For most film based cameras, the viewfinders are optically based and can be used in conjunction with the objective lens of the camera. However, in the digital age of photography, electronic viewfinders are replacing many of the optical viewfinders. Electronic viewfinders offer the ability to add digital overlays, eliminate the parallax error between the viewfinder and the objective optics, and supplement the field of view with digital enhancements to the image in real-time to mimic post-processing as the footage is being captured.
An electronic viewfinder is coupled to the visual system of the user and has associated performance and mechanical specifications. The design form for a viewfinder falls into the eyepiece class, requiring an external aperture stop in which to position the eye. Significant eye clearance (the distance between the eye and the closest feature of the optics) is needed, with a minimum of 10 mm to clear the eyelashes of a user. More comfortable eye clearance is found at around 25 mm or more. Another important consideration that is crucial for comfortable usability is the eyebox diameter. In general, the eye pupil varies from 2 mm in bright sunlight to over 8 mm in the dark. Typically, in a viewer where the eye is illuminated by an electronic display, the human pupil ranges from 2.5 to 4 mm in diameter, depending on the level of light in the image [15]. To allow the pupil to rotate freely and for slight translations of the head without losing the image, the viewfinder is designed with an oversized eyebox diameter of 12 mm. The final image quality will be evaluated over 3 mm subpupils that sample the entirety of the 12 mm eyebox to accurately characterize the system in the way in which it is used.
Whereas an optical viewfinder relays light directly from the scene to the eye, an electronic viewfinder images light generated by a miniature electronic display. There are currently two main competing technologies in the miniature display market – organic light emitting diode (OLED) microdisplays and those based on liquid crystal technologies (LCD and LCoS). The main benefit that OLED microdisplays have over their liquid crystal counterparts is self-illumination. LCoS and LCD microdisplays require an external illuminator, thereby increasing the overall package size. For that reason, a commercially available 1080p OLED microdisplay was chosen for this design study. However, OLED microdisplays do suffer from color uniformity issues for non-normal illumination angles. To mitigate this effect, the system is restricted from having a tilted image plane (OLED) and the system must maintain a high degree of telecentricity (chief ray angle at the image plane). We will further highlight in Section 3.2 how this OLED property is of key consideration in defining a starting point for the final design.
It is uncommon to find an unobscured reflective eyepiece because they generally cannot handle the required aperture and field specifications; hence the majority of eyepiece designs are refractive. However with the use of freeform surfaces, it will be shown that the unobscured reflective system design space opens up to allow for large apertures and fields within a compact form. Design specifications for the viewfinder system are shown in Table 1.
Table 1. Electronic Viewfinder Specifications
3. Viewfinder design process using freeform aberration theory
The aberration correction benefits from freeform surfaces are most readily seen in the design of rotationally nonsymmetric imaging systems. The use of reflective surfaces in an electronic viewfinder allows us to bend the light into a compact package, while eliminating an often limiting aberration of eyepiece-type systems: lateral color. The design process for an unobscured reflective freeform optical system is an iterative procedure with the first step being the development of a feasible first-order starting point based on imaging performance and package size/geometry requirements. After a suitable starting point has been developed, the field behavior of the limiting aberration(s) is determined using FFDs. Once identified, the limiting aberrations are addressed by adding the corresponding freeform correction term to a surface within the optical system, as identified by [12]. The process of identifying the field dependences of the limiting aberrations and implementing the freeform correction term is repeated until either the system specifications are met or no further improvement is possible for the selected configuration, due to the balancing of aberration types. This will be illustrated in detail within this section for an electronic viewfinder.
3.1 Freeform aberration theory
While the treatment of freeform aberration theory found in [12] is beyond the scope of this paper, it is beneficial for the reader to have a basic understanding of the content of that work. Fuerschbach et al. derived and illustrated the aberrations that are imparted into a system as a result of adding a specific Fringe Zernike polynomial shape to a surface. In short, the addition of each Zernike polynomial to a surface adds a field-constant contribution of the aberration associated with the specific Zernike polynomial, along with aberrations with lower-order pupil dependence, with varying field dependences. The theory is valid for surfaces located anywhere within the optical system. The magnitudes of the generated aberrations vary based on their location relative to the aperture stop of the system. Knowing the effect that a freeform surface will have on the aberrations of a system guides the designer to optimally choose how to use freeform surfaces. The aberrations that are generated for Zernike astigmatism (Z5/6), coma (Z7/8), and elliptical coma (Z10/11) surfaces located in an optical system are shown in Fig. 2. Various means of describing a freeform surface have been reported (i.e. orthogonal polynomials [16–19], radial basis functions [20], XY-polynomials [1], NURBS [21]); because our design methodology uses the theory of aberration fields for freeform surfaces to guide how freeform terms may be added in optimization, it requires the use of Zernike polynomials as detailed in Sections 3.2 and 3.3. Thus in this paper, Zernike polynomials are used as the freeform descriptor for the optical surfaces.
Fig. 2 The resulting aberration contributions after adding a Zernike (a) astigmatism, (b) coma, or (c) elliptical coma contribution to a surface within an optical system. The resulting aberrations are (a) field-constant astigmatism, (b) field-linear medial field curvature, field-linear field-asymmetric astigmatism, and field-constant coma, and (c) field-constant elliptical coma and field-linear field-conjugate astigmatism. (Image adapted from [12])
3.2 Starting design choice
Developing a first-order geometry for a reflective system that is unobscured at the full FOV and aperture specifications is a top priority. An investigation of whether one, two, or more mirrors are required to meet specifications was first conducted. The simplest solution is a single reflector placed far enough from the aperture stop and tilted to provide the necessary clearances. However, after a baseline optimization, it is quickly seen that a single freeform surface does not have sufficient degrees of freedom to meet the required performance specifications. The next set of first-order geometries includes two-mirror solutions. To keep the system compact, only positive-positive solutions were considered. Two custom unobscured spherical design forms that support the first-order specifications were optimized and are shown in Fig. 3. Both solutions suffer from comparably large tilt-induced aberrations, such as field-constant astigmatism and coma, as seen in Figs. 4 and 5. Also note that the OLED microdisplays could be tilted to mitigate the focal plane tilt in these starting points, however this degree of freedom was removed in place of satisfying the system telecentricity constraint. The choice between these two geometries is a critical step in the overall design process for the viewfinder system. The field dependences of the astigmatism and defocus/field curvature (FC) are roughly the same for both geometries, but the orientations of the coma are opposite one another. Referencing Fig. 2, it can be seen that when a comatic surface (Z7/8) is added to a system, it adds defocus/FC, astigmatism, and coma in the same orientations as in Fig. 4. This point hints that, for the geometry in Fig. 3(a), adding a comatic contribution can address the defocus/FC, astigmatism, and coma simultaneously, while for the geometry in Fig. 3(b), the astigmatism and coma are present in opposing amounts, only allowing the balancing of these aberrations. For this reason, the geometry in Fig. 3(a) is vastly superior, and will be our starting point for the remainder of the design.
Fig. 3 Two possible unobscured two-mirror geometries for the electronic viewfinder that may serve as a starting point. Note that the OLEDs could be tilted to mitigate the focal plane tilt in these starting points, but that degree of freedom was removed with the system telecentricity constraint.
Fig. 4 Aberration FFDs for the starting point shown in Fig. 3(a) for Zernike (a) defocus/FC, (b) astigmatism, (c) astigmatism after the field-constant component is digitally removed for illustration purposes, and (d) coma. The tilt-induced field-constant astigmatism is dominating in (b). All other aberration contributions not shown here are negligible.
Fig. 5 Aberration FFDs for the starting point shown in Fig. 3(b) for Zernike (a) defocus/FC, (b) astigmatism, (c) astigmatism after the field-constant component is digitally removed for illustration purposes, and (d) coma. The limiting aberrations are similar to those shown in Fig. 4, but the orientation of the coma is rotated 180° due to the alternate rotation of the secondary mirror. All other aberration contributions not shown here are negligible.
3.3 Adding Zernike freeform terms
After choosing a first-order starting point, the next step was to identify the field dependence of the dominant aberration(s). The aberrations for the starting design are shown in Fig. 4. The dominant aberration is astigmatism, which has a near-constant field dependence as a result of the tilted surfaces. As seen in Fig. 2(a), adding an astigmatic term to any surface within the system adds only field-constant astigmatism to the system. Thus, an astigmatic contribution is added to both surfaces within the system, which is then optimized to eliminate the field-constant astigmatism. The resulting aberration contributions after adding the astigmatic contribution are shown in Fig. 6.
Fig. 6 Aberration FFDs for the system after adding an astigmatic contribution to correct the field-constant astigmatism. The plots show Zernike (a) defocus/FC, (b) astigmatism, and (c) coma. Notice the field-constant astigmatism has essentially been optically eliminated, leaving field-linear field asymmetric astigmatism.
The FFDs in Fig. 6 clarify our choice of the starting point. They show the presence of three dominant aberrations with the following field dependences: 1D field-linear medial FC (which can be thought of as a focal plane tilt), field-linear field-asymmetric astigmatism, and field-linear coma, whose node has moved vertically, far outside the field of interest. As a reminder, the focal plane tilt could be removed by tilting the OLED, but the system telecentricity constraints do not allow that. The orientations of the three aberrations are such that adding a comatic contribution to a surface within the system addresses all three dominant aberrations simultaneously. The FFDs after adding the comatic surface are shown in Fig. 7. The RMS wavefront error improved by a factor of 6 after adding a comatic contribution but the system performance does not yet meet the desired specifications, so the design process is continued to seek the limit of what may be achieved with a two-mirror freeform solution.
Fig. 7 Aberration FFDs for the system after optimizing with a comatic contribution on both surfaces. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error. Note the scale decrease of 6x from Fig. 6.
Continuing on with the freeform design procedure, two dominant aberrations are identified in Fig. 7 as field-linear field-conjugate astigmatism (Fig. 7(b)) and nearly field-constant elliptical coma (Fig. 7(d)). These two aberrations can be addressed in tandem through the addition of an elliptical coma contribution to a surface within the system. The residual aberrations after adding the elliptical coma contribution are seen in Fig. 8.
Fig. 8 Aberration FFDs for the system after optimizing with an elliptical coma contribution on both surfaces. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error. Note the ~2x scale decrease from Fig. 7.
It is at this point that the defocus/FC will be addressed, which is now the dominant aberration in the system. By letting the base curvature of each surface become a conic, and by adding a Zernike defocus and spherical aberration contribution, the defocus/FC of the system can be significantly decreased. The residual aberrations after adding the conics and spherical contribution are shown in Fig. 9. Let it be noted that through the addition of freeform Zernike terms, the optical system geometry can be modified to assume surface tilts that better suit the system as the optimization progresses.
Fig. 9 Aberration FFDs for the system after adding conics and spherical aberration contributions to the mirrors. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error.
At this stage in the design and optimization, the dominant aberrations are not overwhelmingly dominated by a single field dependence, as was the case for the previous stages; the numerous field behaviors for each aberration have begun to balance with one another. So, as a final step in the design, Zernike oblique spherical (Z12), fifth-order coma (Z15), and tetrafoil (Z17) contributions are incrementally added to the system to help refine the system and meet the performance requirements. The extra degrees of freedom will also help with other system constraints such as distortion and telecentricity. The final design layout and surface shapes that are the result of the design process are shown in Fig. 10. The final viewfinder aberration FFDs are shown in Fig. 11.
Fig. 10 (left) The final design layout. The two mirrors are in a slightly different orientation than the starting design. (right) The primary and secondary mirror shown with the best-fit manufacturing sphere removed. The maximum departure of the primary and secondary mirrors are 350 µm and 330 µm, respectively.
Fig. 11 Aberration FFDs for the final viewfinder system. The plots show Zernike (a) defocus/FC, (b) astigmatism, (c) coma, (d) elliptical coma, (e) oblique spherical, and (f) RMS wavefront error. The average RMS wavefront error is 0.46 waves. Note the scale decrease of 2.5x from Fig. 9.
4. Evaluation of the final viewfinder
With the design of the electronic viewfinder complete, a set of performance analyses will now be performed. The design of the viewfinder was done with a 12 mm diameter eyebox, but the viewfinder will be used with the eye, which is approximated with a 3 mm diameter pupil. The system is evaluated with a 3 mm pupil in both centered and decentered configurations over the entirety of the 12 mm eyebox. The regions over which the full eyebox is sampled are shown in Fig. 12. The system performs near the diffraction limit for all 3 mm subpupils sampled over the full eyebox. The MTF performance at 40 lp/mm and 50 lp/mm (the Nyquist frequency for the OLED) for the limiting case subpupil is shown in Fig. 13. The distortion of the system is shown in a simulated image of a scene viewed through the viewfinder as calculated by the 2D Image Simulation function in CODE V, shown in Fig. 13. If the distortion of the image is unacceptable for a given application, predistortion of the image shown on the OLED is an available option [22].
Fig. 12 The black outer circle represents the full 12 mm eyebox diameter over which the 3 mm subpupils are sampled to simulate the eye as the aperture stop of the viewfinder. The system is symmetric about the vertical axis, so only fields on one half are sampled. The limiting subpupil is the filled circle.
Fig. 13 (left) MTF FFD results for 0 degree and 90 degree orientations of the object in display space for the limiting case subpupil highlighted in Fig. 12. The columns show 40 lp/mm (80% Nyquist of OLED) and 50 lp/mm (Nyquist of OLED). (right) A simulated image seen through the viewfinder. The 4% distortion is best seen at the bottom corners.
As described in [4] and [23], to accurately describe the performance of the system, it must also be evaluated in the fashion in which it will be used. A visual space evaluation requires the system to be reversed in the optical design software to simulate light traveling from the microdisplay to the eye. The MTF performance for the limiting 3 mm pupil position is shown in Fig. 14 for angular frequencies that correspond to spatial frequencies of 40 lp/mm and 50 lp/mm. The maximum resolvable angular resolution for those with “normal” 20/20 vision is 0.5 lp/arcmin, so because the system has significant contrast at frequencies greater than 0.5, it is nominally eye-limited.
Fig. 14 MTF FFD results for the viewfinder in visual space for 0 degree and 90 degree object orientations for the limiting case subpupil highlighted in Fig. 12. The angular frequencies correspond to the spatial frequencies shown in the MTF FFDs of Fig. 13.
5. Sensitivity Analysis
To evaluate the sensitivity of the final design, a root-sum-square (RSS) sensitivity analysis was performed for the following assembly and surface shape errors: 10 arcminute tip/tilt/clock of the mirrors and OLED, 100 µm despaces and mirror decenters, and 1/2 wave (at 632 nm) peak-to-valley surface figure errors for three errors commonly seen for freeform parts (spherical aberration, astigmatism, and coma). No compensation was used. The analysis was performed for the subpupil with the lowest nominal MTF, shown in Fig. 12. The f-number for the system evaluated with a 3 mm subpupil is f/16, so it is expected that the system will be insensitive to these changes.
The RSS method is composed of individually perturbing each parameter within the system with the maximum tolerance value. The performance of the system is then recorded. After every parameter has been individually perturbed, the RSS is taken of the resulting performance drops to yield a final estimated performance drop of the as-built system. This method allows the designer to identify and address the most sensitive system tolerances. The set of tolerances chosen for this system are quite loose, and yet, the performance drop is within an acceptable range. This result documents the idea that freeform systems are not as sensitive as one might assume. Table 2 shows the MTF impact for each tolerance in the system at the on-axis field point and the most sensitive off-axis point for both the 0° and 90° object orientation.
Table 2. Percent MTF drop for each tolerance at 35 lp/mm, (λ = 632 nm).
6. Conclusion
The recently developed aberration fields of freeform optics enhances the design process of freeform optical systems by providing the insight necessary to first develop the optimal starting point and then efficiently optimize the freeform surfaces of the system using Zernike polynomial surfaces. The freeform surfaces allow the optical system to overcome the significant aberration induced by tilting the mirrors into an unobscured package. An all-reflective, freeform, 25° full FOV electronic viewfinder with a 12 mm eyebox was successfully designed and design principles that are based on the aberration theory for freeform surfaces were demonstrated. The analyses showed near diffraction limited performance for the 3 mm subpupils and low sensitivity to manufacturing tolerances, leaving margin for error in the fabrication of the freeform surfaces and the alignment of the system.
Acknowledgments
This work was supported by the NSF I/UCRC Center for Freeform Optics (CeFO) (IIP-1338877 and IIP-1338898). We thank Julius Muschaweck and Matthias Pesch for stimulating discussions about this research. We also thank the CeFO collaborative consortium as well as Synopsys, Inc. for the student license of CODE V®.
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