Optimization of high-performance monocentric lenses

Transcription

1 Optimization of ig-performance monocentric lenses Igor Stamenov,* Ilya Agurok, and Josep E. Ford Department of Electrical and Computer Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla, California , USA *Corresponding autor: Received 23 September 2013; accepted 15 October 2013; posted 28 October 2013 (Doc. ID ); publised 25 November 2013 Te recent application of monocentric lenses for panoramic ig-resolution digital imagers raises te question of te acievable performance limits of tis lens structure and of tecniques for design optimization to approac tese limits. Tis paper defines te important regions of te design space of moderate complexity monocentric lenses and describes systematic and global optimization algoritms for te design of monocentric objective lenses of various focal lengts, apertures, and spectral bandwidts. We demonstrate te trade-off between spectral band, F-number and lens complexity, and provide design examples of monocentric lenses for specific applications Optical Society of America OCIS codes: ( ) Lens system design; ( ) Imaging systems; ( ) Systems design. ttp://dx.doi.org/ /ao Introduction Monocentric imaging lenses, wic are constrained to ave only sperical surfaces centered on a single point of symmetry, can produce a ig-resolution image on a sperical image surface. Since ig-resolution sperical detectors are not currently available, in practical application tis image surface is optically transferred onto multiple conventional focal planes. Tis can be done by relay troug multiple adjacent sets of secondary optics, as in monocentric multiscale imagers [1, 3]. Alternately, it can be done via imaging fiber bundles wit curved input and flat output faces, as in te monocentric fiber-coupled imagers [4 9]. Tese successful demonstrations motivate a more systematic exploration of te capabilities of te monocentric imaging lens. In a lens wit centered sperical or emisperical surfaces, off-axis aberrations of coma and astigmatism are cancelled [10], but we need to correct sperical and cromatic aberration and teir combination sperocromatism. Reducing sperocromatism is difficult, especially wit large apertures. But despite X/13/ $15.00/ Optical Society of America te monocentric constraint, and even wit a small number of degrees of freedom, it is possible to obtain a number of useful designs [3,4,8,11]. In a previous paper [8], we reported te general aberration analysis of two-glass symmetric (2GS) monocentric lenses and results of our 2GS global searc algoritm applied for a specific example, a 12 mm focal lengt, F/1.7, 120 field of view lens operating in nm visible waveband. Te algoritm described identified te optimum diffraction limited design [Fig. 1(a)] and a number of additional families of ig-performing solutions. However, if we substantially increase te lens spectrum, ligt collection, or te scale, even after repeated 2GS global searc, we will not acieve desired performance [Figs. 1(b) 1(d)]. Te 2GS monocentric arcitecture reaces its limits. Acieving a similar level of performance wit tese extended operating specifications demands more complex monocentric lens arcitectures, wit more degrees of freedom. Tis is especially true wen increasing more tan one of tese performance metrics. In tis paper, we sow metods and algoritms for advanced monocentric lens design. We categorize te monocentric lens design space, provide procedures for optimum and near-optimum lens design 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8287

2 Fig. 1. MTF performance curves sowing te limits of te globally optimized 2GS monocentric geometries. Te examples are derived from an initial ig-performing lens (a), wic is pused to improve spectral bandwidt (b), numerical aperture (c), or focal lengt (d) [using a 3 scale cange to te illustration]. For eac, te resolution of te two-glass structure drops well below te diffraction limit, indicating a need for greater complexity. wit complexity and performance trade-off considerations, and in Section 5 provide specific lens designs for selected applications. 2. Options for Improving Monocentric Lenses A. Review of Monocentric Lens Arcitectures Te simplest monocentric lens arcitecture is a simple glass ball [12,13] wit inset aperture stop. Historically, te more common approac was an acromatic 2GS arcitecture, used by Sutton in 1856, Baker in 1942 [14], and more recently by Brady and co-workers [1,3] and Ford and co-workers [2,15]. A more complex tree-glass symmetric structure (3GS) wit tird-order aberration analysis was designed and fabricated by Oakley [16] for a panoramic sperical retroreflector. However, ig order aberrations at large apertures were not systematically corrected. Our goal was to pus te performance of te existing lens and identify te limit of wat monocentric lenses can or cannot do. We started by using commercial lens design software to explore te monocentric lens design space by a systematic increase of degrees of freedom in te system, wile maintaining te monocentricity constraint, to identify te major configurations, wic sowed te most promise. We constrained te focal lengt and did a lens optimization for all options wit a given number of degrees of freedom (i.e., glass coice, surface radius) and compared performance to te diffraction limit. Glass as an optical material as at least two description parameters, te index of refraction, and te Abbe number. To model te dispersion over a broader spectral range would require an expression wit even more free parameters. But since we don t ave te ability to create a glass wit arbitrary index and dispersion, te coice of an existing glass material represents only a single degree of freedom. We use teir accurate models described by Sellmeier, Extended, or Scott glass model formulas. Figure 2 summarizes te result, sowing 100 different geometries and te seven preferred design arcitectures (drawn wit a larger scale). Tose preferred arcitectures were labeled as 1GS: One-glass symmetric wit 1 degree of freedom (DOF) 2GS: Two-glass symmetric wit 3 DOF 3GS: Tree-glass symmetric wit 5 DOF 3GA-7: Tree-glass asymmetric wit air gap and 7 DOF 4GA-8: Four-glass asymmetric wit air gap and 8DOF 4GA-9: Four-glass asymmetric wit air gap and 9DOF 5GA-10: Five-glass asymmetric wit air gap and 10 DOF. Te 1GS, 2GS, 3GS, and 4GA-8 geometries were cosen for rigorous analysis and investigation, because tey offered te best performance for teir structural complexity. Te simplest 1GS geometry is a symmetric glass ball, wit only one degree of freedom (1 DOF). Wen Fig. 2. Monocentric lens design space sowing glass only (upper alf) and glass wit air gap (lower alf) regions divided by te seven preferred design arcitectures in between APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013

3 te desired monocentric system is being designed, focal-lengt input constrains one of te radii, so te coice of glass in tis structure remains as te single variable. Since tere is no cromatism correction, tis arcitecture is suitable mostly for monocromatic imagers wit a relatively large F-number. In air, if we allow tis geometry to become asymmetric, or increase degrees of freedom to two, te optimizer will converge back to te original 1GS structure as depicted in te upper alf of Fig. 2. A similar outcome results if we introduce an air gap and pus up to te maximum of six degrees of freedom (lower part of Fig. 2). Te next logical step was to make te acromatic lens wit an additional glass, wic yields te 2GS geometry wit tree degrees of freedom. As in 1GS geometry, optimization after lens splitting and introducing an air gap will converge back to te simpler 2GS geometry, wile taking te upper glass modification only pat on te cart by allowing all radii to vary will land in a 5DOF two-glass arcitecture tat performs substantially te same as te simplest 2GS structure. Adding te tird and fourt glass in te monocentric structure and breaking te symmetry goes deep into te upper glass modification only region of te design space and offers only marginal improvement over 2GS and 3GS structures, not justifying te cost of manufacture. Terefore, simple symmetry breaking and glass adding is not productive. Te upper alf of te diagram is only partially populated if we allow more glasses and more degrees of freedom, arcitecture will essentially converge to some variant of te Luneburg lens solution [17]. On te oter and, starting from 5DOF structure wit tree glasses and introducing an air gap also doesn t appear to elp, as we continue along te lower part of Fig. 2 (glass wit air gap pat) up to te 6t degree of freedom. Ten just a simple step over to te 7 degrees of freedom 3GA-7 arcitecture gives a substantial increase in performance, as sown wit red arrow in Fig. 2. Furter derivatives 4GA-8, 4GA-9, and 5GA-10 just keep up wit te same trend. Out of tese asymmetric structures wit an air gap, 4GA-8 is te most attractive one to pursue wit an addition of 5GA-10 for larger scale lenses were te maximum glass slab size plays an important role. Looking at tis compreensive monocentric lens design space cart, an interesting fact is tat simply adding te degrees of freedom at some point does not elp. Tis is somewat counter-intuitive. For te two, four, and six degrees of freedom cases, no preferred monocentric lens structure exists. All symmetry breaking attempts in tis 6 DOF area inevitably converge back to te symmetric structures wen te lens is designed for te use in air. A sligt cange to tis rule applies only wen te lens as a different medium in object and image space (e.g., an immersed lens), were a 4 DOF two-glass structure wit symmetric core becomes te preferred design. Our next goal was to find specific igperformance designs. To do tis we developed global searc algoritms for symmetric geometries, and systematic searc metods for te asymmetric geometries wit an air gap, as described in te following. B. Review of Monocentric Lens Design Metods Trougout te exploration of monocentric lens design space, several metods and optimization algoritms were developed. In previous work [8], we presented a global optimization algoritm for te 2GS arcitecture. Now a similar approac was used in one-glass (1GS) and tree-glass (3GS) symmetric arcitectures, and a similar global searc algoritm was developed. Since all tese global optimization routines are essentially brute force calculations (for all possible glass combinations), wit furter increase in te number of degrees of freedom te cost of computing became proibitive. Terefore, we developed systematic searc metods. All te metods use spectral band, focal lengt, and F-number as an input for te desired system, and a predefined pool of commercially available glasses. Tese included te Scott, Oara, Hoya, Sumita catalogs as well as CAF2 and fused silica, totaling 604 different materials available as of April Hikari, CDGM, and NHG manufacturers were not used because almost all of teir glasses represent duplicate replacements of te glasses already included. Te optimization metods used, in order of increasing complexity and computation time, were 1GS global optimization algoritm (seconds to complete) 2GS global optimization algoritm (minutes to complete) 3GS global optimization algoritm (days to complete) 2GS seeded Hammer searc (ours to days) 4GA-8 arcitecture 5-D near global searc (up to 3 weeks). Preliminary results of tese metods were presented in [18] but will be described in more detail ere. Global optimization algoritms for 1GS, 2GS, and 3GS arcitectures are multitreaded exact ray trace routines implemented in MATLAB. Tese ceck all possible glass coices (604 for 1GS, 364,816 for 2GS, and more tan 220 million combinations for 3GS geometry). Tey were executed on PC workstations wit two Intel 3.1 GHz Xeon E W or four Intel 2.7 GHz Xeon E Sandy Bridge based processors (16 32 CPU cores systems). Te 3GS global optimization algoritm was also rewritten and tested on Kepler based NVIDIA K20 Tesla and K5000 Quadro GPU cards, wit speed improvements on te order of 70, effectively cutting down te computing time required from days to ours. Te 2GS seeded Hammer searc approac used glass combinations of te top 2GS candidates obtained troug global searc, ten imported in 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8289

4 Fig. 3. Optimization of preferred monocentric lens geometries. were ΔS is longitudinal aberration of te ray of eigt i. Also, we modified te polycromatic mean square wavefront deformation calculation for an increased number of wavelengts and increased te Zernike polynomials expansion to te nint order. Te system polycromatic mean square wavefront deformation ΔΦ 2 became ZEMAX. Te lens symmetry was broken and an air gap manually introduced. Additional glass layers were added one by one, as sown in Fig. 3, wit optimization at eac step. In practice, te 2GS geometry was optimized by uman assistance troug ZEMAX Hammer searc to 3GA- 7 and ten to 4GA-8 arcitecture or even more complex ones, if needed. Te most complicated design approac, wic was guaranteed to give te best result for te 4GA-8 geometry, was a five-dimensional near global searc algoritm, also implemented in MATLAB. Like te seeded Hammer searc, te 5D optimization algoritm starts wit te core identified as te best 2GS candidates, and ten te algoritm tries all combinations for te additional tree glasses used in te 4GA-8 geometry. Because of five-dimensional optimization space complexity, tis algoritm requires up to 3 weeks to complete running continuously on te 32CPU core workstation. We used tis algoritm at te end of te lens design procedure to determine te absolute best candidate for manufacture and test. 3. Advanced Design Algoritms and Results A. Improved 2GS Global Searc Using 5 Wavelengts In te previous paper [8] we reported a general aberration analysis of 2GS monocentric lenses, and applied a global searc algoritm to identify te best design for a visible waveband monocentric lens. To generate a ranked list of all lens candidates, tis algoritm used a tree-step optimization for eac possible two-glass combination: minimization of tirdorder Seidel sperical and longitudinal cromatism aberrations, exact ray trace for multiple ray eigts at te central wavelengt, and te calculation of polycromatic mean square wavefront deformation. Te algoritm described was sufficiently accurate for te visible (potograpic) spectral range, were te glass dispersion curve is approximately linear. However, to look for solutions in an extended waveband, we modified te existing 3λ algoritm by replacing te first two steps by exact ray trace wit five equally spaced wavelengts over te desired spectrum. Te modified exact ray trace cost function became ( Q X5 X 3 Abs ΔS i ; λ m m 1 X3 i 1 ) X Abs ΔS j ; λ m ΔS k ; λ m ; (1) j 1 k j ΔΦ X 5 i 1 C new 20 λ i 2 C 40 λ i C 80 λ i 2 9 C 100 λ i 2 11 C 60 λ i 2 7 : (2) We ten used te improved 2GS global searc 5λ algoritm wit te updated glass catalog to look again for te optimal designs for f 12 mm F/ nm camera and a longer focal lengt lens needed for te AWARE 2 Gigapixel imager, wit f 70 mm, F/3.5, and a nm spectrum. For te f 12 mm case, te previously optimal top family of solutions remained on top wile quite a few intermediate (but still inferior) families were generated (Table 1). For simplicity, since tere are many similar glasses in te catalogs, a number of glasses tat ave a refraction index witin 0.03 and an Abbe number witin 2 of te glasses sown we considered as replacement glasses and omitted from te table. On te oter and, te longer f 70 mm lens benefited significantly from te increased number of materials, and we identified two better candidate families tan previously reported: K-VC82/P-LAF37/ S-BAH11/K-LasFN10 wit a CAF2 core [as sown in Fig. 4(a)] and M-LAF81/MP-LAF81/ L-LAM69/S-LAH60 wit K-GFK68 core [as sown in Fig. 4(b)]. Te monocentric 2GS global searc generates a full list of ranked solutions, wic is wy a global searc algoritm is more powerful tan simply doing guided ammer/global searces in a commercial optical design software like ZEMAX or CODEV. From a ranked list, te lens designer can quickly coose te specific designs subject to specific constraints suc as lens volume, differential termal expansion, or glass material availability. B. Tree-Glass Symmetric Global Searc After reacing te 2GS arcitecture limits, we explored te 3GS arcitecture sown in Fig. 5. Similar to 2GS geometry, from te first order principles, te focal lengt is given by [8,16] 1 f 2 1 1n2 2r2 r 1 1n2 1n3 2r3 1n3 1n4 (3) 8290 APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013

5 Table 1. Updated List of Top Solutions for te SCENICC F/1.7 f 12 mm nm 120 Monocentric Lens a Fast Exact Ray Tracing [mm] ZEMAX Optim. Radii [mm] No. Outer Glass Internal Glass R1 R2 ΔΦ 2 R1 R2 MTF at 200 lp mm 1 S-LAH79 K-LaSFn9, TAF5, S-LAH TAFD55 K-LaFK50, S-YGH52, M-TAC (LASF35) 3 N-LASF46A/B M(C)-TAF1, TAF5, K-LaFK50(T), (TAFD25, L-LAH86) S-LAH59, K-LaSFn9, S-LAH65(V) 4 L-NBH54 K-LaFn9, S-LAM K-GIR79 (LAH80, N-LASF9) K-LaFK50T, M(C)-TAF1, N-LAF21, K-LaSFn16, TAF TAFD40 M-TAFD305, L-LAH85V, L-LAH S-LAH79 M-TAF101, N-LAF21, K-LaSFn16, TAF4, M-TAF1, TAC4, K-LaKn12 8 K-PSFn5 N-LASF45(HT), S-LAM TAFD40 N-LAF2, K-LaF2, LAF2, S-LAM2, K-LaFn11, S-LAM61 10 LASF35 (S-LAH79) K-LaK9, K-LaK12, N-LAK12, S-LAL12 ; K-VC80, K-LaK13, P-LAK35, L-LAL13, S-LAL N-LASF46A/B (TAFD25, L-LAH86) N-LAK12, K-LaK9, N-LAK12, S-LAL12, LAC12, L-LAL12 a Prescriptions sown pertain to glass combinations, marked in bold For eac cosen glass combination, one radius is a function of te oter two radii and te predefined focal lengt target value. From te real ray-trace geometrical equations we obtained eigts. To allow for extended spectral bands and material dispersion curves, a ray trace is done for nine equally spaced wavelengts inside te spectrum of interest. In te 3GS geometry tere is around 220 million glass combinations, and te optimization OE n io : (4) sin 2 arcsin r 1 arcsin r 1 n 2 arcsin r 2 n 2 arcsin r 2 n 3 arcsin r 3 n 3 arcsin r 3 n 4 Te longitudinal aberration for te ray wit input eigt i is given by ΔS i OE i f (5) Finally, we constructed te following merit function for 3GS geometry optimization: problem is inerently two-dimensional. In an attempt to reduce te computing time to reasonable limits, we identified and made use of an interesting fact about 3GS geometries. In te two-dimensional optimization space of te 3GS monocentric system, if te glass coice is viable, areas of minimum merit Q X9 X9 X 8 i 1 j 1 X 8 i 1 j 1 p j Abs ΔS j ; λ i p j Abs ΔS p j f NA;λ i ; (6) were f is te focal lengt of te lens, NA te numerical aperture, and p 1; 0.9; 0.8; 0.7; 0.6; 0.5; 0.4; 0.3 are te pupil zones used to calculate eigt ray Fig. 4. New f 70 mm AWARE 2 2GS candidates. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8291

6 Fig. 5. Monocentric tree-glass symmetric (3GS) arcitecture. function (ig performance) look like a long and nearly linear ravine [Fig. 6(a)]. So it is possible to fix te radius of te second glass sell at two points wit some reasonable values, cross te ravine at two r 2 levels to get its orientation, and ten trade te two-dimensional optimization problem for one-dimensional track along te ravine. Tis increased te computational efficiency and made it possible for te global searc to run in on a ig-performance workstation (4 2.7 Hz Intel Xeon E5-4650). Because te ravine is substantially flat at te bottom, we ad te freedom to coose te second radius in 3GS system, and tis was elpful in avoiding excessively tin sell solutions, wic are impractical to fabricate. Figure 6(b) sows te comparison of 2GS and 3GS top candidates for te 12 mm F/1.7 monocentric lens operating in te extended visible ( nm) waveband. Bot solutions are strong apocromats [Fig. 6(c)], but unfortunately te 3GS geometry offered only modest performance improvement in MTF and te spot size. A similar result is observed wen te global 3GS searc is applied in all scenarios for te f 12 mm imager lens variants discussed in Section 1 of tis paper. 3GS global searc generates a number of ig-ranked solutions tat ave nearly identical performance, and it is difficult to say wic one is te absolute best performer. Some solutions ave sligtly better MTF but worse RMS spot size, and vice versa. Te candidates sown in Fig. 7 are cosen by MTF performance. An interesting fact about te 3GS geometry is tat all good solutions are always derivatives from te good 2GS candidates. In oter words, glass core materials of te top candidates identified in 2GS global searc also form top 3GS solutions. Tat kind of beavior was observed in all design scenarios. Terefore te quick track to global 3GS solutions may be te exploration of all glass combinations, constrained only by te limited number of materials for te core. Tis effectively reduces te CPU computing time from days to ours or, in te case of GPU computing, from ours to minutes. After a number of global 3GS searces and comparisons, we finally concluded tat over te scale of apertures and waveband parameters we considered ere, te 3GS arcitecture does not offer a significant performance improvement over te 2GS arcitecture. C. Seeded Hammer Optimization For te desired monocentric lens specification, as a start, a 2GS global searc was performed and te full list of ranked candidates was created. Ten, te multiple top candidate prescriptions were imported to ZEMAX and manual lens splitting and air gap introduction were performed: first guiding te candidate optimization to 3GA-7 structure, and ten to 4GA-8 structure. All glass materials were substitution variables except te core, and witin ours (sometimes even minutes) Hammer searc would find a useful solution. We must empasize tat in order for tat to appen, te most important ting is a good starting core material for te given design. Witout 2GS global searc algoritm input, bot ZEMAX and CODEV may ave a ard time converging to te best solutions if te starting design core material is not close to te ideal, especially for te low F-number cases. Te reason for tat was te sape of multiple local minimums in te monocentric design space as discussed in te previous paper [8]. Optimized 4GA-8 structures troug te ZEMAX Hammer optimization seeded from top 2GS candidates of te modified 12 mm imager specification lenses are sown in Fig. 8. Te original lens [Fig. 8(a)] was substantially Fig. 6. Top 12 mm F/ nm 3GS monocentric lens candidate (a) optimization space, (b) MTF comparison curves wit top 2GS candidate, and (c) apocromatic saped focal sift curve APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013

7 Fig. 7. MTF performance comparison of globally optimized 2GS and 3GS monocentric lenses for extension of te original lens specifications. Te plots sow only on-axis MTF, to allow comparison of 2GS and 3GS arcitectures. Fig. 8. MTF performance curves of te 4GA-8 lens geometries derived from te original lens specifications troug seeded Hammer optimization. diffraction-limited in bot te 2GS and 3GS geometry, so wit te original design specification only a sligt improvement is seen. For te oter tree cases, owever, te more complex lens acieved substantial performance improvement. Hammer optimization is not global, and even iger performance designs may exist, so te 4GA-8 arcitecture is a promising coice for igperformance lenses. Tis lens is sufficiently complex tat an exaustive and truly global optimization is impractical, but in te following section we describe a five-dimensional 4GA-8 monocentric arcitecture optimizer to identify near-global lens designs. D. Five-Dimensional 4GA-8 Near Global Optimization A useful solution is to break te front/rear symmetry and introduce an asymmetric air gap between te crown and flint glass core [5,19]. Introducing suc an air gap is a common metod used for control of sperocromatism [20 22]. Tis approac yields te four-glass air gap asymmetric geometry, wic improves performance on extended spectral bands, larger apertures, and longer focal lengt systems. Te four-glass wit air gap (4GA-8) lens arcitecture is sown in Fig. 9. Attempts to optimize te four-glass arcitecture wit ZEMAX software sows tat te result of optimization strongly depends on te initial starting point position. Some results obtained from very different starting points in te radii space sowed good image quality but oters were trapped in lower quality pockets. Suc beavior of te commercial lens design software indicates tat te optimization space Fig. 9. Four-glass asymmetric wit air gap (4GA-8) monocentric lens arcitecture. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8293

8 of te 4GA-8 monocentric lenses as some specific features tat must be investigated and special optimization algoritms to be developed. As for te 2GS and 3GS arcitectures, for lens quality evaluation we used fast exact monocentric lens ray tracing. For te input ray at eigt it gives te value for te lengt OG (all radii values assumed positive): CMOS or CCD sensor, we used te waveband 0.4 to 1.0 micrometers, divided into eigt equal segments at nine wavelengt values. Tis criterion demonstrated a good correlation wit modulation transfer function (MTF) for all types of monocentric lenses operating in extended wavebands. OG n sin arcsin R 1 arcsin R 2 n 2 arcsin R 2 n 4 arcsin R 1 n 2 2 arcsin arcsin R 4 arcsin R 4 n 4 arcsin R 5 arcsin R 5 n 5 arcsin R 6 arcsin R 2 n 3 o R 6 n 5 7 Te longitudinal aberration ΔS i for tis ray will be ΔS i OG i f; (8) were f is te focal lengt. To form te optimization criterion, te results of fast exact ray tracing wit Eqs. (7) to(8) were used. Te entrance eigts of tese rays are i NA f p i ; (9) were p i is an array of reduced rays eigts at te pupil. Te array is defined as p : (10) For te optimization criterion C, te following sum was used: C X9 X 9 i 1 j 1 p j ΔS i 2 ΔS 9 ; λ λ 1 ΔS 9 ; λ 9 2 j ΔS 3 ; λ 1 ΔS 3 ; λ 9 2 ΔS 1 ; λ 1 ΔS 1 ; λ 9 2 ; (11) were λ j is te wavelengt in micrometers used for weigting. Te first term of te criterion C equation is a sum of squared values proportional to lateral aberrations and te following tree members are squared cromatic longitudinal aberrations differences at te pupil reduced rays eigts 1, 0.88, and Te longitudinal cromatic difference at te reduced pupil eigt 0.05 is similar to te classical cromatic focus sift. Pupil points wit reduced pupil eigt 1 and 0.88 are critical for te sperocromatism correction. For optimization of any monocentric lens operating in an extended waveband, we used nine wavelengts. For example, for criterion calculations for monocentric lenses designed to operate wit a front-illuminated silicon Te starting point for te 4GA-8 systematic searc is to make use of te core from multiple 2GS top candidates as seeds for furter optimization. Te most promising glass K-GFK68 was cosen as a basic core glass for te systematic solution searc, and ten te oter glasses were replaced in all possible combinations. For eac glass combination, te searc of minimum of te criterion C [Eq. (11)] was performed and te optimized system was found. Wit te cosen glass combination we ave five radii to optimize. In fact tere are seven radii in te optical sceme, including te image surface radius (Fig. 9), but te tird radius is equal and opposite to te second radius because te use of te central ball lens, and te image surface radius must matc te focal lengt. Investigation of te criterion C beavior sowed te multi-extremum nature of te function being optimized. But in our case, on top of tis problem, we ave a number of linear combinations between optimization parameters, or in oter words, lines and surfaces in te optimization space over wic te criterion does not cange or canges very slowly. Te optimization process is stuck somewere in tese ravines depending on te starting point position. Suc areas are multidimensional ravines or saddle type stationary areas. For eac glass combination, te minimum value of te criterion C is located at different positions in te radii space, but all glass combinations still ave te caracteristic general sape of solutions in te 5D radii space. For every glass combination, te contour surfaces wit te constant values of criterion C around te minimums appear as tin, pancakesaped volumes in te 5D radii space. Tese tin pancake volumes are pierced wit a net of saddle type ravines, and are connected over te main ravine. Applying conventional optimization metods results in slow convergence to a solution trapped in te saddle type ravines, rater tan te global minimum [23,24]. Te beavior of te gradient metod in suc cases is illustrated in Fig. 10 (adapted from [24]). Figure 10(a) sows te gradient metod beavior in te general case of te normal minimum sape. Te gradient descent direction at any step is directed ortogonally to te criterion contour line 8294 APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013

9 Fig. 10. Gradient descent metod applied for normal minimum and degraded minimum sape. and straigt descent is continued up to te point wen it reaces anoter lower value contour line for wic te direction of descent would be tangential. Te process converges quickly to te minimum in just a few steps. In te case of a degraded (stretced) minimum wit te strong linear dependence between optimization parameters, as sown in Fig. 10(b), te gradient metod begins to oscillate. In [25], te metod of conjugated gradients was proposed. It was sown tat apexes of segmented lines in te gradient metod are located on te lines sowing te direction to te minimum and after several steps we can create tese lines by using te least square metod. Te move along tese lines will facilitate a fast descent to te minimum. Wile te metod of conjugated gradients can elp in a number of degraded minimum cases, our situation is more sopisticated. Te minimum volume as te sape of a tin pancake in te 5D space, wit te walls close to being parallel, suc tat gradient descent segments X i 0 Xi 1 0 and X i 1 0 X i 2 0 will practically coincide. Te points X i 0, Xi 2 0, X i 4 0 will be located so close to eac oter tat we will not be able to reliably connect tem wit te line. Moreover, all te points inside te five-dimensional tin pancake minimum area are saddle-like points. At every point we ave Hesse matrix [23,26] aving one nearly zero negative eigenvalue, demonstrating a strong linear dependence between te first (R1) and last (R6) radii, wile oter eigenvalues will be strongly positive. Te saddle type nature of te area of te minimum solution is anoter reason tat conventional optimization metods are likely to be trapped at different points inside te pancake, were te specific endpoint depends sensitively on te initial starting point of te optimization. In tis situation, even te metod of conjugated gradients fails. Te optimization of our lens arcitecture required te development of special metods, wic we will illustrate using te example of a lens wit te following glass combination: P-LASF47, K-GFK68, K-LASFN6, and N-KZFS11. Tis glass combination demonstrated sufficiently good performance during our searc for te optimal solution and was cosen as an example to demonstrate te optimization procedure. Our searc for te near minimum begins wit a gradient descent [23] to te closest local minimum from te average radii solution for tis arcitecture (Fig. 9). Te optimization of tis glass combination begins from te average radii combination at point M0 7.0; 2.9; 2.9; 4.2; 4.5; 7.8; 12.0, in millimeters. Te value of te criterion C [Eq. (11)] at tis point is Te local gradient descent metod quickly arrives to te point inside pancake area wit radii array M ; ; ; ; ; , wic again are sown in millimeters, and criterion value at tis point is C Te contour lines graps of te criterion C [Eq. (11)] in te plane section of radii R1-R6 of te five-dimensional space is sown in Fig. 11. Te minimum area is a tin long strip, wic is te section of a five-dimensional tin pancake. Te minimum point M1 is sown wit information box on Fig. 11(a). We used te Hesse matrix eigenvectors [26] to find te direction of te strip. Te five eigenvalues of te Hesse matrix at tis point are , , , 25.44, and Note tat radius number tree does not participate in te optimization because it is equal to te negative value of te second radius. Te two eigenvectors aving smallest eigenvalues are E ; ; ; ; and E ; ; ; ; , were te eigenvector direction cosines are related to te radii R1, R2, R4, R5, and R6, respectively. Te section of criterion C sows tat te main ravine is related to te eigenvector E1. We will name tis ravine as te main virtual ravine (Fig. 12). Knowing tat gradient metods are descending in te direction ortogonal to te contours of constant criterion C value, it is not surprising tat gradient metods from any initial point come toward te main virtual ravine but to different locations over tis ravine depending on te location of te initial point. Te 3D grap of te criterion C wit dependence on te first and last radii is sown in Fig. 11(b). Te value of te criterion function C in te direction ortogonal to te ravine quickly reaces te value of 0.25, wit Fig. 11. Optimization criterion ravine of minimums (projection onto 3D space). 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8295

10 Fig. 12. space. Optimization procedure inside te 4GA-8 optimization radii space step size as small as 0.02 mm. We will use te main ravine as an entrance area into te pancake space. Te ravine is located very close to te straigt line associated wit te eigenvector E1 (te black dotted line in Fig. 12). We travel over tis straigt line tat is defined by te first eigenvector direction at te point of te minimum M1 and make a number of local optimizations, wic will quickly come to te actual lowest point of te ravines. Following over te direction of eigenvector E1 we made 17 steps wit 0.05 increments in te radii space (8 steps in te direction of lower first radius and 8 steps in te opposite direction) and, making te gradient descent from eac point, obtained te array of minimums. Table 2 sows te central 10 minimums (ravine bottom points) of tis scan wit minimum M1 over te main ravine at place number 9. Table 2 sows tat minimums are located over a deep and sligtly curved ravine, wit a strong linear dependence between te first and last radii. Te body of te pancake saped minimum is located over te 3D spere inside te 5D space, and tis spere is ortogonal to te ravine, as sown in Fig. 11(b). Te directions of te minimum increment of te criterion C at eac bottom point over te ravine is a direction of te second eigenvector wile te first one is still directed over te ravine. Te tunnels into te pancake (black dased lines in Fig. 12) [26] are located over te directions of second eigenvectors. Tese vectors are ortogonal to te ravines (blue das-dot lines in Fig. 12). To find te point of te criterion C minimum we crossed te ravine structure inside te pancake area of solutions from te point M1 in te direction of te second eigenvector E2. Ten we initiated local gradient descents wit te step of mm. Values of C over tis line after local gradient descents are (point M1), ten [ , , , , , , , , ]. Te ravine wit te minimum value of criterion C is at te point M2, aving C and te array of radii is M ; ; ; ; ; Ten we made 17 steps along eigenvector E1 of tis ravine wit te sort gradient descent. Te values of C over ravine are [ , , , , , , , , , , , , , , , , ]. Te minimum value of C is at te point M3 wit array of radii M ; ; ; ; ; We performed tis wole operation in cycles until te step wen te minimum is located at te initial point of te last cycle. Te wole optimization pat is sown by te solid red line in Fig. 12. Values of C around M3 are [ , , , , , , ], were te new minimum point M4 as C , and array of radii M ; ; ; ; ; Te next step along tis new ravine associated wit te point M4 gives te point M5 at one step from M4 wit C Te array of radii at M5 is M ; ; ; ; ; Te next crossing of te ravines did not succeed and te minimum C point remained at te point M5, indicating tat we ad approaced te limit of tis process. Te next step was to substitute te array of radii at te point M5 wit C into ZEMAX software, were we obtained te MTF value of 0.54 at 200 lp mm frequency. After tat we used te standard Zemax process for a local optimization of te optical prescription to obtain te maximum MTF. Te final results are sown in Table 3. For te MTF optimized sceme MTF at 200 lp mm is It is sligtly better tan we ad at te optimum point of criterion C (Eq. 11). Te construction Table 2. Array of Local Minimums Over te Main Ravine in 4GA-8 Optimization Space. a C initial r r r r r r C final a Radii sown pertain to te areas at te bottom of te ravine reaced after optimization APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013

11 Table 3. Optical Prescription of te nm F/1.7 f 12 mm MC Lens Example Solution A 5.63 g Radius Tickness Glass Semi-Diameter P-LASF K-GFK STO Infinity K-GFK K-LASFN N-KZFS IMA of a low burden computer criterion from te results of raytracing, wic will perfectly correlate wit MTF performance, is still an open problem [24,27]. However, we consider tat our criterion is in good correlation wit MTF, allowing us to sort te results of te searc for 4GA-8 arcitecture. Te optimization process from te different initial point MR0 aving R1 7.5 mm sows anoter solution inside te neigboring pancake area on te rigt (Fig. 12), wit te value of C Te optimized MTF at frequency 200 lp m for tis solution is Te optical prescription is sown in te Table 4. Similarly, te next optimization process from te initial point ML0 aving R1 6.5 mm sows anoter solution inside te neigboring pancake on te left wit C Te optimized MTF at frequency 200 lp m for tis solution is Te optical prescription is sown in te Table 5. In te global Table 4. Optical Prescription of te nm F/1.7 f 12 mm MC Lens Example Solution B 5.71 g Radius Tickness Glass Semi-Diameter P-LASF K-GFK STO Infinity K-GFK K-LASFN N-KZFS IMA Table 5. Optical Prescription of te nm F/1.7 f 12 mm MC Lens Example Solution C 5.72 g Radius Tickness Glass Semi-Diameter P-LASF K-GFK STO Infinity K-GFK K-LASFN N-KZFS IMA searc among tese tree feasible solutions we prefer te solutions of first type sown in Table 3, as tey ave te smallest weigt. Te procedure described above was applied on all oter glass combinations, and our near-global searc resulted in 350 top solutions tat are grouped in seven distinctive families (Table 6). Glasses are considered replacement glasses if teir index of refraction is witin 0.03 range and Abbe number in 3 range of glasses sown in te table. Te example solution discussed before, sown in Tables 3 5, belongs to te first family of solutions. Figure 13 sows te optical layout of te lens and MTF curves of te top solution from te first family and compares it wit te seeded Hammer result from Fig. 8(b). Te prescription of tis near-global optimum solution is sown in Table 7. Upon inspection of te full solutions list, te seeded Hammer solution was located wit C near-global searc criterion value and obviously far outside te top solutions families. Our near global 5D searc improved te MTF performance at 200 lp mm over te seeded Hammer solution by a 16% margin. Figure 14 sows graps of sensitivities of te front and back-illuminated silicon sensors [28]. Our next goal was to modify te nm 4GA-8 solution from Table 3 to operate wit frontilluminated silicon sensor. We constructed ZEMAX merit function as a function keeping at minimum radii of point spread functions at all nine wavelengts (operators REAR), maximizing MTF at frequencies 100, 160, and 200 lp mm and keeping te focal lengt at 12 mm (operator EFFL). Substitution of te wavelengt weigts for te front-illuminated silicon sensor and quick reoptimization in ZEMAX gave te optical prescription sown in Table 8. Te image quality is practically diffraction limited. Te MTF of te lens is sown in te Fig. 15. At te 200 lp mm te lens as 90% level of diffraction limited resolution. Back-illuminated silicon sensors are sensitive to as low as 200 nm wavelengt. We found tat acromatization in nm waveband is out of ability of te 4GA-8 arcitecture at tis scale and aperture. In order to avoid te use of expensive coatings we decided to cut off te UV spectrum by using te mounting meniscus made from te Scott GG435 absorptive glass. Using an additional mounting meniscus at te curved image surface is optional and as little impact on imaging system performance. Te optical prescription of te monocentric lens operating wit te back-illuminated silicon sensor is sown in Table 9. GG435 color glass refraction indices were measured wit Filmetrix F10-RT refractometer. Te MTF and te layout of te lens are sown in te Fig. 16. Te image quality is close to diffraction limited. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8297

12 Table 6. Families of Solutions for F/ mm Monocentric 4GA nm Lens Obtained Troug Near-Global Searc Family 1st Glass 2nd Glass (Core) 3rd Glass 4t Glass (Meniscus) Near-Global Searc Criterion MTF at 200 lp mm I P-LASF50 K-GFK68 TAF1 E-ADF II NBFD11 K-GFK68 TAF3 KZFS III L-LAM72 K-GFK68 TAF1 S-NBH IV TAF2 K-GFK68 P-LASF50 N-KZFS V L-LAH83 K-GFK68 NBF1 KZFS VI TAFD30 K-GFK68 P-LASF50 N-KZFS VII P-LASF51 K-GFK68 S-LAH58 N-KZFS Fig. 14. sensor. Spectral response of front and back-illuminated silicon Fig. 13. MTF curves for 12 mm, F/1.7, nm lenses obtained troug seeded Hammer searc and near global fivedimensional optimization (sown on layout). Bot lenses ave te core glass K-GFK68 wit a very ig coefficient of termal expansion, TCE 12.9, wile surrounding glasses ave low TCE coefficients. For example, te front-illuminated silicon sensor lens sown in Table 8: P-LASF47 glass as TCE 6.04 and K-LASFN6 glass as TCE 5.9. Normally te TCE difference less tan 1.5 for cemented surfaces can be recommended for outdoor optics [29]. Recently Norland Products Inc. offered extremely low psi modulus NOA 76 [30] optical cement, wic can be used for glass pairs wit suc ig CTE differences. ZEMAX termal modeling of te scemes sown in Tables 8 and 9 wit a 10 micrometer tick layer of NOA 76 optical cement for differential termal expansion sows tat te lenses can operate in a wide temperature range of 20 Cto 50 C witout image quality degradation. Only a 0.02 mm back focal lengt adjustment is required. Since monocentric lenses were originally designed to be used wit refocusing [8], tis procedure does not pose a problem. Finally, if we ask ourselves wy te 4GA-8 arcitecture as suc capabilities for acieving te ig performance monocentric designs, te answer lays Table 7. Optical Prescription of te nm F/1.7 f 12 mm Monocentric Near Global Solution Table 8. Optical Prescription of te nm F/1.7 f 12 mm Monocentric Lens Operating wit Front-Illuminated Silicon Sensor Radius Tickness Glass Semi-Diameter P-LASF K-GFK STO Infinity K-GFK TAF E-ADF IMA Radius Tickness Glass Semi-Diameter P-LASF K-GFK STO Infinity K-GFK K-LASFN N-KZFS IMA APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013

13 Fig. 15. MTF curves for 12 mm, F/1.7 lens operating wit nm front-illuminated silicon sensor sensitivity spectrum. Fig. 16. MTF curves for 12 mm, F/1.7 lens operating wit nm back-illuminated silicon sensor sensitivity spectrum. Table 9. Optical Prescription of te nm F/1.7 f 12 mm Monocentric Lens Operating wit Back-Illuminated Silicon Sensor Radius Tickness Glass Semi-Diameter NBFD K-GFK STO Infinity K-GFK K-LASFN KZFS GG IMA in te asymmetry and te presence of an air-gap. To support tat claim, on Fig. 17 longitudinal aberrations over te pupil for 3GS and 4GA-8 arcitectures are sown. Symmetric designs like 2GS and 3GS always ave te sape of te longitudinal aberrations, as in Fig. 17(a). Tis single bending curve, as te aperture grows, cannot be controlled, so te sperocromatism and zonal sperical aberration become dominant and limit te performance. On te oter and, presence of an air gap and broken symmetry elp correcting tose aberrations and adds an additional bending to te curve as in Fig. 17(b). 4. Lens Complexity and Performance Trade-off To explore te maximum acievable performance of te monocentric lens geometries, we began wit te design constraint of a 12 mm focal lengt 120 FOV imager and te visible waveband of nm and optimized te lens design to increase te aperture as muc as possible, subject to a predefined performance metric. Tis performance constraint was to require tat te MTF was at least 70% of te Fig. 17. Longitudinal aberrations of te (a) top 3GS and (b) top 4GA-8 arcitecture 12 mm F/1.7 monocentric lenses for nm spectral band. diffraction limit at 200 lp mm (te igest spatial frequency needed for Nyquist sampling of a 2.5 micrometer pitc Scott 24AS optical fiber faceplate) wile simultaneously requiring tat te RMS spot size radius ad to be less tan 1.5 te Airy disc radius (wic maintained MTF at lower spatial frequencies). Wile tis metric is somewat arbitrary, te trends of te results are indicative of a wide range of related performance metrics, as applied to te available degrees of freedom. Te results are sown in Fig. 18. Ill cosen geometries, labeled in blue in Fig. 18 (1G-2, 2GAS, 3GA-6), converge to simpler ones and do not enable an increase in aperture size. Te geometries labeled in black (1GS, 2GS, 3GS) sow results obtained by our global optimization algoritm, wile geometries labeled in red (3GA-7, 4GA8) are te result of te seeded Hammer optimization, starting from top 2GS candidates as seed designs. 1 December 2013 / Vol. 52, No. 34 / APPLIED OPTICS 8299

14 Fig. 18. Monocentric lens geometries optimization beavior for two different scales operating in nm spectral range. Adding te second glass is elpful in controlling te cromatic aberration, so tere is a large increase in acievable F-number in moving from geometry 1GS to 2GS. Tis is equivalent to going from a singlet lens to a cemented doublet acromat. Adding te tird glass in te 3GS geometry gives marginal cromatic aberrations improvement over 2GS, wereas te oter degrees of freedom (specifically, 2GAS wit 4 DOF, and 3GA-6 wit 6 DOF) provide no improvement. Breaking te symmetry and introducing an air gap wit te 7 DOF and 8 DOF arcitectures allows us to furter increase te aperture, and still meet te desired MTF performance. Similar beavior is observed for te longer 112 mm focal lengt lens, wic is also sown in Fig. 18. Te final step was to explore te design space for four different focal lengt scales {f 12 mm (te SCENICC program lens scale), 40 mm, 70 mm (AWARE2 program [1]) and 112 mm (AWARE10 program [31])}, and at eac scale look at te maximum aperture for visible, extended visible, and visible-nir spectral wavebands, subject to te MTF performance constraint described above. As before, we used te global optimization for te two-glass and tree-glass symmetric systems (2GS, 3GS). For te tree- and four-glass air gap candidates (3GA-7, 4GA-8), te combination of our systematic fift-dimensional optimizer wit ZEMAX ammer optimization was used. Te entire set of results is summarized in Fig. 19. Te clear trend is maintained over all tree spectral bands and it sows te necessity of lens structure complexity Fig. 19. Monocentric objective lens performance trade-off for different scales and tree spectral bands ( nm, nm, nm) APPLIED OPTICS / Vol. 52, No. 34 / 1 December 2013