Orthogonally Coupled Geometrical Constraints in Optics Design Jyrki Kimmel

Kimmel Journal of the European Optical Society-Rapid Publications Journal of the European Optical (2019) 15:25 https://doi.org/10.1186/s41476-019-0121-4 Society-Rapid Publications

RESEARCH Open Access Orthogonally coupled geometrical constraints in optics design Jyrki Kimmel

Abstract Modern optical multi-camera systems require integrating many camera modules in a small volume. A new space-saving concept for such imaging systems is presented, based on intersecting optical paths that utilize one or more common elements for the respective optical paths. The principles for the optimization for such systems is examined, providing the theory for geometric optimization constraints. These principles can become useful in designing e. g. spatially challenging 360-degree imaging systems for surveillance and consumer applications. Keywords: Optics design, Design theory, Orthogonal constraints, Coupled constraints, 360-degree imaging

Introduction optical function simultaneously for both optical paths. Imaging refractive optical systems are most often de- The easiest way to think of this type of design is to inter- signed by using sequential raytracing software, and the sect the optical axes orthogonally to each other at the lo- underlying principle in design is to optimize the optical cation of the common element. surfaces to a merit function that is tailored to the speci- Previously, common-element based imaging arrays have fication of the optical system under consideration [1]. been designed for gigapixel imaging for instance, in the While this approach is proven and quite straightforward, DARPA (Defense Advanced Research Projects Agency) there are design problems that cannot be satisfied with SCENICC (Soldier Centric Imaging via Computational simple barrel-based optics where the optimization is Cameras) program [3–5]. In the AWARE (Advanced Wide done with respect to one optical axis. For example, field of view Architectures for Image Reconstruction and multi-camera imaging systems for extreme wideangle Exploitation) program within SCENICC, a monocentric photography aim to provide a full 360-degree spatial im- multicamera array was designed to provide 2-gigapixel [4] aging capability, combining the image space of several and 100-gigapixel [5] imagery by arranging a multicamera optical barrel-sensor systems by software means, such as array at the back focal region behind a single or dual elem- stitching [2]. These systems are typically more than 15 ent spherical lens system. For this approach to work for all cm in diameter, where it is assumed that individual en- the individual cameras, all surfaces in the spherical lens sys- trance pupils of the lenses are ca. 60 mm or more apart, tem were concentric, forming, in effect, a multilayer ball. approximating or slightly exceeding the human interpu- The cameras each had a relay optic to convey a partially pillary distance (IPD) between the optical axes of the overlapping image field on the array of individual image lenses [2]. This matching is important for achieving real- sensors. Free-space or fiber-optic relay optics were used for istic three-dimensional (3-D) imaging capability, to each individual camera to provide focusing on the flat present the imaged content to the user with virtual real- image sensor [5]. ity viewing devices. The immediate effect of applying a common element In order to save space, a design framework based on for two optical paths in the same design is that the intersecting optical axes is proposed. The underlying optimization needs to dimension the common element principle in this design is to utilize, for both optical axes, in such a way that the propagation of the full field needs one or more common optical elements to provide an to be able to pass through this element, in both orthogonal directions. The overall effect of designing systems Correspondence: [email protected] with intersecting optical axes is that the optimization pa- Enterprise & Industrial Automation Laboratory, Nokia Bell Laboratories, P. O. rameters for each axis as well as their tolerances are Box 785, 33101 Tampere, Finland

© The Author(s). 2019 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 2 of 9

singly or multiply coupled to each other. This article ex- smaller semidiameter than the second surface of the com- amines the taxonomy and optimization principles of mon element, whether it is an air gap or an optical glass such systems and discusses practical design workflow as- element. Taking first the case of converging marginal rays, pects of optical dual-axis systems with orthogonally the simplest system utilizes a central air gap in the design coupled constraints. To the Author’s knowledge, this is (Fig. 1a). If the air gap is replaced with an optical material, the first report where the theoretical limit to minimize we arrive at the design of Fig. 1b). Considering these de- the dimensions of an optical design, where the optical signs, but replacing the marginal ray with a diverging one, paths intersect orthogonally within a common element, in Fig. 1c) the central element is again an air space, and in is described. Fig. 1d), the central element is an optical material. These four proto designs constitute the basis of the taxonomy Taxonomy of optical designs for orthogonally for finding the geometrical constraints to minimize the coupled optical systems distance between the first surface of the common element Limiting the approach of this study to orthogonally and the intersecting orthogonal optical axis. Further de- coupled dual optics where the optical design of both inter- sign considerations arise with regard to the positioning of secting axes is identical, a taxonomy can be developed, the stop in the optical paths which is shown in Fig. 2 that taking into account the common aspects applied in the presents the taxonomy tree. The taxonomy tree shows all design (Fig. 1). One major differentiator in the designs is theoretically possible configurations, however, some of the sign of the marginal ray angle at the leading surface of these may not be feasible with regard to making practic- the common element. The marginal ray can be converging able imaging optics. In the following, the stop is only or diverging, i. e. the first surface can have a larger or regarded as any arbitrary surface within the design.

Fig. 1 Four basic cases of the orthogonally coupled optics design: a Converging marginal ray, air as central element; b Converging marginal ray, glass as central element; c Diverging marginal ray, air as central element; d Diverging marginal ray, glass as central element Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 3 of 9

Fig. 2 Taxonomy of orthogonally coupled optics design

There are differences as to how the optimization Constraints for the common central element constraints imposed on thedesignwillbemadein The constraints that apply in the optical design of or- order to force the optimization process to realize a thogonally coupled systems relate to the dimensions of given topology of the taxonomy tree. The common the central element whether it is air (Fig. 1a and c) or an feature, however, is the dimensioning of the central optical material (Fig. 1b and d). Furthermore, the con- element, whether it is made of air or an optical mater- straints limiting the geometry can be defined in terms of ial. The topology outlined in the taxonomy will deter- the signs of the leading and trailing surfaces of the com- mine the mathematical and the theoretical limit to mon central element and the sign of the marginal ray which these designs can be minimized in size. In angle β, as is conceptually shown in Fig. 3. From Fig. 3, addition to constraining the central, common element, we can see that for each optical path arrangement utiliz- also limiting constraints for the surrounding elements ing a central, common element, the leading surface S in the complete design will need to be imposed. can either be positive (convex toward the left) or negative (convex toward the right). Similarly, for the trailing surface T, the same conditions regarding the signs of the Mathematical principles in the constraint surface apply. For the leading surface S, the marginal ray definition for the dual orthogonal design leaving S at angle β can either be converging (β <0)or In the following treatment, it is assumed that in the dual diverging (β > 0). The limiting calculations must be done orthogonal design, both optical functions, in each of the for both the leading surfaces S and S′, as well as the orthogonal optical paths, are identical. The treatment is trailing surfaces T and T’, where the primed symbols also limited to spherical surfaces constituting the com- refer to the orthogonal axis and its respective surfaces, mon, central element. It is of course possible to design as either surface, when placed too close, can potentially space-saving systems where the optical designs for either partially block the passage of light from the orthogonal axis are different, or where the intersecting axes are light train. In total, there are eight conditions for which non-orthogonally oriented. Similar mathematics can be the mathematical expressions limiting the thickness of developed for these cases, using the principles outlined the central element need to be developed (see Table 1). in this article. First, the mathematical expressions or the Table 1 presents the taxonomy of Fig. 1 as a table with constraints of the element are developed (Constraints abbreviations that signify the eight main cases that guide for the common central element section) and then, the the placement of the common central element, now tak- constraints of the surrounding elements are described ing into account also the signs of the curvatures of the (Constraints for the surrounding elements section). leading and trailing surfaces. Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 4 of 9

is that the near edges of the surfaces S and S′ do not intersect:

eS > hS ð2Þ

where hS is the marginal ray height at the surface S (or, clear semidiameter of surface S). When β > 0, respectively, the far edge of surface S′ must not intersect with the marginal ray. From trigonometry, it is easy to obtain: Fig. 3 The basic considerations for orthogonally coupled optics design. Identical considerations apply in the orthogonal direction eS > hS Â ðÞ1 þ tanβ =ðÞþ1− tanβ sS ð3Þ (not shown here) qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Where sS ¼ rS − rS − hS is the sag of surface S and rS is The mathematical framework is the same, irrespective the radius of the surface S. of whether the central element is an air space or an op- At the trailing surface, conditions expressed by (1) and tical material. The key principle is in ensuring that the (2) are applied conversely. For the diverging case (β > 0), surfaces of the orthogonal train of light passage do not interfere with the propagation of light of the other main eT > hT ð4Þ direction. Therefore, in Table 1, the surfaces S and S′ as β well as T and T’ are designated to be either positive or and for the converging case ( < 0), negative, following the conventions used in optical simu- eT > hT Â ðÞ1 þ tanβ =ðÞþ1− tanβ sT ð5Þ lation software [6]. The constraints also only differ with regard to the sign of the marginal ray angle (converging where the quantities hT and sT refer to the marginal ray rays or diverging rays). height and sag, respectively, of surface T and T’. It is helpful to think of the central point in the de- The calculation for the conditions where the lead- sign as the intersection of the orthogonal optical axes. ing surface S is negative leads to a more complicated This intersection is therefore located within the com- expression, which can be solved as an abstract trig- mon element. As it does not represent a real surface onometric problem (see Fig. 4). Mathematically, this in the design, it can be incorporated in optimization can be expressed in terms of hS,themarginalray as a “dummy surface” D, which in Zemax OpticStudio height at surface S and rS, the radius of the surface is a surface with no effect on light propagation [6]. S, as well as the marginal ray angle β of the surface Therefore, in the following calculations, the distance t S. These quantities are obtained in sequential optical after the surface S and S′ (the distance between S simulation from the optimization of the leading sur- and T as well as between S′ and T’) will be the sum face. Starting with the following trigonometric iden- of the distances eS and eT to and from the intersect- tities from Fig. 4,wheretheangleα is the angle ing optical axis (dummy surface D)(subscriptsS and defined by the height hS of the marginal ray leaving T refer to the leading surface S and the trailing sur- surface S (point q)andtheradiusrS of surface S face T, respectively): (α =arcsin (hS/rS)). The fundamental equations for line A,lineB perpendicular to line A,andcircleS′ are: t ¼ eS þ eT ð1Þ y ¼ tanβ x þ b; b ¼ hS À tanβ dS ðÞðfor Line A 6Þ From Fig. 3, it can be seen that there are conditions y ¼ −1= tanβ x þ f; ð7Þ where the surfaces never can overlap the light path of f ¼ ðÞ1−1= tanβ cS ðÞfor Line B the orthogonal light train. These occur when the leading surface S is positive, and the trailing surface T is negative 2 2 2 rS ¼ ðÞx À cS þ ðÞy À cS ðÞðfor Circle S’ 8Þ (cases C + - and D + - in Table 1). In these cases, if the 2 2 2 marginal ray is converging (β < 0), an adequate condition denoting m = tanβ and dS = rS – hS , x and y can be

Table 1 Classification of constraint conditions for the central element. The constraints arise from the sign of the leading and trailing surfaces (positive or negative) as well as the sign of the marginal ray leaving the leading surface (converging or diverging) Leading surface / Trailing surface Positive Negative Positive Converging (C++) Diverging (D++) Converging (C − +) Diverging (D − +) Negative Converging (C + −) Diverging (D + −) Converging (C −−) Diverging (D −−) Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 5 of 9

o o o o eliminated, and for cS, we obtain the solution of a quad- divided by zero: β ≠ 0 , β <45 , β > −45 .Atβ =0 ,we ratic equation (for full derivation, see Appendix): arrive at the trivial solution eS = hS. Note that the above pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −B Æ B2 À 4AK derivation leading to (13) applies both to the converging cS ¼ ð9Þ and diverging marginal rays. 2A Conversely, again, a similar calculation can be applied where the coefficients are to the positive sign trailing surfaces to obtain eT (the “ ” D A ¼ m4 À 2m3 þ m2−2m þ 1 ð10Þ thickness [7] of the dummy surface to the trailing ÂÃÀÁÀÁ surface T), substituting the relevant quantities hT (mar- 2 2 B ¼ 2 m−m m dS−mhS þ ðÞm−1 ðÞhS−mdS ginal ray height at the surface T), dT (distance of the T ð11Þ marginal ray intersection point with surface in the horizontal direction), and rT (radius of surface T) for the 4 2 3 2 K ¼ m ds − 2m dShS − 2mdShS − ds ð12Þ marginal ray angle β. For designs dimensioning the central element by (13), first and foremost, the optimization constrains the thick- S To obtain the distance from surface to the orthog- ness of the central element to equal its diameter adding onal optical axis, we identify for our final result from the sag of the surfaces (the sign of the sag included). Fig. 4: Here, the marginal ray of the field corresponding to the largest field of view (FOV) of the lens design needs to be eS ¼ cS À rS ð13Þ able to pass through the central element without even In the calculations, there arise conditions that exclude tangentially hitting the side surfaces of the element. some angles due to the geometry of the problem (Fig. 4) Therefore, the clear semidiameter of the central element, as well as due to that the sides of the equation cannot be as well as its thickness, may need to be larger than is

Fig. 4 Lens design constraints for the central element after preceding surface S. The orthogonal optical axes are denoted by the dot-dashed lines, and they also represent the dummy surfaces D (vertical axis) and D’ (horizontal axis). The marginal ray leaves the leading surface S at point q, at the marginal angle β. The closest the orthogonal surface can get to the intersection of the two orthogonal axes is limited by line A which is the tangent of surface S′, defined by the marginal angle β at point p (grazing incidence for the marginal ray). Line B is perpendicular to line A and is defined by points o and p Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 6 of 9

necessary for conventional single-barrel designs, to pass element can be solved using the principles presented in this marginal ray. Eq. (13), and rules for dimensioning the other elements can be found in Eqs. (14)and(15). A flowchart for the Constraints for the surrounding elements design process can then be developed, depending on the If the maximum diameter of the first surface in the lens set of initial requirements and specifications, as shown is specified, several dimensions for the design can be cal- in Fig. 5. culated. Often, the maximum optical track length is also As the flowchart shows, the design first starts with the defined. For instance, the common central element can fundamental specifications for the optical design. Some be placed at half the maximum track length, and the of the geometrical considerations can then be directly upper limit of the semidiameters of all elements can derived from the specified values for the lens diameter then be calculated. and optical track length. Notably, the numerical aperture If the central element is the only common optical part and viewing angle as well as focal length of the objective in the dual orthogonal design, the positioning of this will not play a role in specifying the properties of the central element is not constrained by the surrounding central element. Assuming that the most suitable appli- surfaces, provided that the semidiameters of the surfaces cation of optics is in wideangle multicamera systems, n of the preceding elements prior to the intersection of hyperhemispherical optical designs commonly apply the orthogonal axes are not, respectively, larger than telecentric back focal systems [7]. Therefore, the com- their distance en subtracting their sag sn, from the mon element can very well be placed behind the aper- central dummy surface D of the common element. For ture stop, making it simple to design the central element each preceding element, therefore: using the constraint calculations of constraints for the common central element and constraints for the h <¼ e −s ð Þ n n n 14 surrounding elements sections. Indeed, the main design consideration in the lens specification will be the choice where hn is the semidiameter of surface n (note that the for the starting design. Systems with many surfaces in sign of the sag as defined in Zemax OpticStudio [6]is the middle and toward the back focal range of the design important). The center of the common element or the will probably not be suitable to apply the orthogonal corresponding air space (i. e. the intersection of the op- crossed-path design outlined in this article. tical axes) must therefore be placed at least the distance The second main consideration in the design spe- hn from any and all preceding elements. cification concerns the image sensor to be applied in The center of the common element is also a constraint the design. There are many design drivers that de- for the optimization of the optical elements placed after pend on the application of the system to be de- the common element in the optical design, limiting the signed. Some applications aim at a high image semidiameter of each subsequent element to be quality with low distortions, and others are opti- hn <¼ en þ sn ð15Þ mized with regard to high efficiency in light throughput and even near infrared performance. where the same sign convention as in Eq. (14) is used Sensors that suit these applications must be selected, for the sag sn. and the optical design subjected to the requirements It should be noted that the constraining Eqs. (13, 14 imposed by the sensor. and 15) apply for both axes in the dual orthogonal de- Once the specification is defined, the semidiameter sign, simultaneously. Therefore, only one optical path upper limit for the preceding surfaces (before the needs to be solved, as the optical designs will be identi- common element) are written out as geometric ex- cal in both directions, independently of each other, only pressions in the merit function, for each surface using constrained by the geometry of the orthogonal design. (14). Chip zone width [6] must be taken into account All parameters required for the dual-orthogonal design in physically realistic design for each complete elem- originate from either one of these orthogonal axes, as all ent so as not to make the lens large enough to inter- single elements and optical groups in the design are sect with the corresponding element in the identical, along both optical paths. orthogonal path. After this, the common element is dimensioned according to the principles presented in Discussion Chapter 3 (Eq. 13), giving us the thickness of surface The optical designs developed in this article that allow S (to the dummy surface at the intersection of the or- optical paths to intersect present additional degrees of thogonal optical paths). The semidiameter of the difficulty arising from the coupled constraints affecting dummy, central surface is calculated by the Zemax both of the respective orthogonal optical paths simultan- OpticStudio optimization routine, as the marginal ray eously. The problem of dimensioning the central angle β stays the same when passing through the Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 7 of 9

Fig. 5 Flowchart for lens design with orthogonally coupled constraints dummy surface. This calculation is written as a rC ¼ tCD ð16Þ Zemax Programming Language (ZPL) routine in the Lens Data Editor (for optimization in Zemax Optic- where rC refers to the radius of the common surface C, Studio [6]). Finally, the maximum semidiameters of and tCD refers to the track length from the surface C to the trailing elements (those after the common elem- the dummy surface D, i. e. the radius of the common ent) are written out in the merit function, similarly to surface C must equal its distance from the intersection how the preceding elements were limited in size by of the two orthogonal paths. This approach has been Eq. (15). already used in the design of the monocentric multiscale The optimization itself must make use of the common imagers [4, 5], where the common spherical element element and its lower thickness limit. As this will be a surfaces are all concentric. This new constraint ties the new constraint in the design, most likely, additional sur- optical system tightly together with little room for toler- faces must be used to balance the optical performance ancing. A prospective application for this approach with a similar design without a constraining common might be a hyperhemispherical camera system, incorpor- element. ating an outermost, joint spherical lens surface as the If more than one common elements are included in first surface of the design. Such a system with three the design, they involve additional surfaces jointly uti- dual-camera modules is conceptually presented in Fig. 6. lized by both of the orthogonal optical paths in the sys- In the system of Fig. 6, orthogonally arranged imaging tem to be designed. Let us name one of these surfaces C. paths cross at a common element within each of the Then, an additional constraint is imposed on this com- three dual-camera modules. In addition, a joint outer mon surface C: surface concentric with the central, common element Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 8 of 9

has been designed as the outermost surface of the dual- Further work will involve realistic optical designs using camera module. Placing these modules 120 degrees apart the principles presented in this work, as well as develop- provides a 360-degree camera view of the environment, ing the theory for tolerancing these orthogonally coupled with a possibility of viewing the scene in three dimen- designs. The ZPL routines will also be developed, in sions, as the camera views overlap to provide the right- order to properly optimize the designs under orthogon- eye and left-eye views from all directions. ally coupled geometrical constraints.

Appendix Conclusions For the full derivation of the trigonometric solution to The principles outlined in this paper present a way to de- the smallest distance the orthogonally oriented optical sign space-saving optical systems that may make it pos- surfaces S and S’ can be placed, refer to Fig. 4 in the sible to reduce the size of, for instance, multicamera body of this article. systems for security, surveillance, and virtual reality 3-D The problem can be abstracted by: “Find the distance imaging, making these systems better accessible to the eS for which line A, intersecting circle S at point q =(dS, consumer. Previous efforts to design military multiscale hS) at angle β tangentially intersects the circle S’ at point imagers [3–5] relied on dense packing of the imager array p. The center of circle S is at (0, 0), and the center of cir- S’ o r e r e r [4], resulting in significant space savings compared to cle is at point =(S + S, S q+ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiS), where S is the ra- hemispherical imagers composed of dense-packed individ- 2 dius of both S and S’,anddS ¼ h − r2.” ual cameras with no common elements [4]. No theoretical S S limits for placing the optical elements in relation to the From sequential optical simulation, the known quan- r S common element were given, as the imager array size was tities are the radius S of circle (surface) , the marginal Sh β limited by the packing constraints of placing the cameras ray height at surface S, and the marginal ray angle , S as close to each other as spatially possible [4, 5]. For the since these will be solved for surface before the first time, in this research, the theoretical limit to optimization proceeds any further. minimize the space between the first and second surface From Fig. 4, we can identify in a common, central optical element is reported. cS ¼ rS þ eS ð17Þ In optical simulation, the mathematical expressions for the constraints can be incorporated in the merit function and from the fundamental equations of a line and as hard limits in the lens data editor [6](inZemax OpticStudio as Zemax Programming Language (ZPL) y ¼ tanβ x þ b; ð18Þ routines). b ¼ hS À tanβ dS ðÞfor Line A

y ¼ −1= tanβ x þ f; ð19Þ f ¼ ðÞ1 þ 1= tanβ cS ðÞfor Line B

(note that line B is perpendicular to line A,andβ ≠ 0) In addition, we have the equation for a circle

2 2 2 rS ¼ ðÞx À cS þ ðÞy À cS ðÞðfor Circle S’ 20Þ

Let us denote tanβ = m. At point q, for line A:

hS ¼ mdS þ b ð21Þ

from which, solving for b:

b ¼ hS À mdS ð22Þ

Likewise, at point o, for line B:

cS ¼ −cS=m þ f ð23Þ Fig. 6 Three-dimensional representation of a speculative 360-degree multi-camera solution providing a three-dimensional view. This and solving for f: design utilizes three symmetrically positioned identical dual-path modules with orthogonally intersecting optical axes and a central f ¼ ðÞþ =m cS ð Þ common element, as well as one joint outer surface. Some internal 1 1 24 components are also shown in the Figure The relations (20-23) give us for line A: Kimmel Journal of the European Optical Society-Rapid Publications (2019) 15:25 Page 9 of 9

o y ¼ mx þ hS À mdS ð25Þ obvious that eS = hS, and in this problem, β > -45 , since the distance eT from the dummy surface D to the trailing B And for Line : surface T cannot be negative; also, β <45o, since the

y ¼ −x=m þ ðÞ1 þ 1=m cS ð26Þ light path from the orthogonal path cannot pass through surface S. We also have the equation for circle S’ (20). x Acknowledgements Now, we have three equations and three unknowns , The Author is thankful for M. Sc. Peter Eskolin (Nokia Solutions and Networks) y, and cS. Solving first for x at point p (intersection of for the three-dimensional model of Fig. 6. lines A and B on circle S’): Authors’ contributions The Author is the sole contributor to this work. The author read and mx þ hS À mdS ¼ −x=m þ ðÞ1 þ 1=m cS ð27Þ approved the final manuscript.

from which Authors’ information ÂÃÀÁ Jyrki Kimmel is a Principal Researcher at Nokia Bell Laboratories in Tampere, x ¼ ðÞm þ c þ m2d −mh = m2 þ ð Þ Finland. He holds a DSc (Tech.) degree from Tampere University of 1 S S S 1 28 Technology. He is active in the European Optical Society as a Program Committee Member of Topical Meeting 3 for the AGM and a member of the Substituting x to the equation for line A (18)we Industrial Advisory Committee. He is also a member of the Society for obtain Information Display (SID) where he has served in committee and board ÂÃÀÁ ÀÁ positions, as well as of IEEE. His research interests include new technologies 2 3 2 2 for displays for mobile and wearable applications as well as optics design, y ¼ m þ m cS þ m dS−m hS þ m þ 1 ðÞhS−mdS imaging and display technologies for the emerging mixed reality field. Dr. ð29Þ Kimmel has several peer-refereed articles, book chapters, conference papers, and patents in these disciplines. Substituting x from (28) as well as y from (30) which c Funding both contain S to (20) we get: No funding beyond my salary from Nokia has been applied for or used. 2 ðÞm þ 1 cS þ m2dS−mhS−ðÞm2 þ 1 cS Availability of data and materials m2 þ 1 The mathematical derivation is available by e-mail. 2 ðÞm2 þ m cS þ m3dS−m2hS þ ðÞm2 þ 1 ðÞhS−mdS −ðÞm þ 1 cS þ Competing interests m2 þ 1 The author declares that he has no competing interests. 2 −rS ¼ 0 Received: 11 March 2019 Accepted: 4 October 2019 ð30Þ

2 2 2 2 Recognizing r S = h S + d S and that m + 1 > 0 for all References o β > -45 , and solving for cS from the numerator of the 1. Fischer, R.E., et al.: Optical System Design, 2nd edn. SPIE Press, McGraw-Hill Education (2008) resulting expression, we finally obtain a quadratic 2. Kimmel, J.S., Baldwin, A.R., Rantanen, V.P.: Optics for virtual reality equation: applications. In: Proc. European Optical Society Annual Meeting 2016 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (EOSAM 2016), pp. 82–83 (2016) −B Æ B2 À 4AK 3. SCENICC, https://www.fbo.gov/ (2019). Accessed 23 Aug 2019 cS ¼ ð31Þ 4. Tremblay, E.J., Marks, D.L., Brady, D.J., Ford, J.E.: Design and scaling of 2A monocentric multiscale imagers. Appl. Opt. 51(20), 4691–4702 (2012) 5. Stamenov, I., Agurok, I.P., Ford, J.E.: Optimization of two-glass monocentric lenses for compact panoramic imagers: general aberration analysis and where the coefficients are specific designs. Appl. Opt. 51(31), 7648–76661 (2012) 6. Zemax OpticStudio 18.9, https://www.zemax.com/products/opticstudio A ¼ m4 À m3 þ m2− m þ ð Þ (2019). Accessed 19 Jan 2019 2 2 1 32 7. Martin, C.B.: Design issues of a hyperfield fisheye lens. In: Proc. SPIE 5524, ÂÃÀÁÀÁ Novel Optical Systems Design and Optimization VII (2004) 2 2 B ¼ 2 m−m m dS−mhS þ ðÞm−1 ðÞhS−mdS ð33Þ Publisher’sNote Springer Nature remains neutral with regard to jurisdictional claims in 4 2 3 2 K ¼ m ds − 2m dShS − 2mdShS − ds ð34Þ published maps and institutional affiliations. To obtain the distance from surface S to the orthogonal optical axis, we identify from Fig. 4:

eS ¼ cS À rS ð35Þ which is our solution to the problem. Note that the positive root of (32) is the only meaningful solution to our problem. In the trivial case where β =0o,itis