Design of Wide Angle and Large Aperture Optical System with Inner Focus for Compact System Camera Applications

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Article Design of Wide Angle and Large Aperture Optical System with Inner Focus for Compact System Camera Applications

Hojong Choi 1 and Jaemyung Ryu 2,*

1 Department of Medical IT Convergence Engineering, Kumoh National Institute of Technology, Gumi-daero 39253, Korea; [email protected] 2 Department of Optical Engineering, Kumoh National Institute of Technology, Gumi-daero 39253, Korea * Correspondence: [email protected]; Tel.: +82-54-478-7782

Received: 19 September 2019; Accepted: 20 December 2019; Published: 25 December 2019

Abstract: Conventionally, a bright, very wide-angle optical system is designed as a floating type optical system that moves two or more lens groups composed of multiple lens in order to focus accurately. These have been widely used as phase detection auto focus (AF) methods within conventional digital single-lens reflex (DSLR) cameras. However, a phase detection AF optical system cannot be used when recording motion pictures. In contrast, a compact system camera (CSC) performs AF by the contrast method, where a stepper motor is used as the driving source for moving the optical lens. Nonetheless, to ensure that the focusing lens is lighter, these stepper motors should not have high torque and AF must be possible by moving only one lens. Yet, when focusing is performed with only one lens, aberration change due to focusing lens movement is magnified. Therefore, a very wide-angle optical system comprised of a half-angle of view more than 40 degrees and F of 1/4 has not been developed. Here, a very wide-angle optical system was designed with high resolving power that enables high speed AF, even in contrast mode, by moving only one lens while minimizing aberration change.

Keywords: inner focus; optical system; compact system camera; interchangeable lens

1. Introduction “Mirrorless” is a familiar term for general users and consumers of cameras. Its official name is a compact system camera (CSC) and is also known as a mirrorless interchangeable lens camera (MILC) [1]. The CSC is a form of the digital single lens reflex (DSLR) in which the reflective mirror and the optical view finder are removed [2]. As a result, the volume and weight are dramatically reduced when compared to conventional DSLR type cameras. Further, no mirror shock and black out are caused by DSLR because there is no reflection mirror movement inside the camera [3]. As CSCs did not have an auto focus (AF) sensor at their introduction, the camera essentially used contrast detection auto focus (contrast AF) [4]. Hence, AF speed was significantly slower when compared to contrast AF. The phase difference AF system calculates the difference between the reference phase and the current phase and determines the amount of movement required to achieve optimum focus, while the contrast AF moves the lens group by small amounts to achieve the same [5]. As the method determines the contrast by the image sensor and finds the optimum focus, AF was not rapid. However, the latest technological enhancements to improve AF speed, such as hybrid AF using the phase contrast AF method and contrast AF method simultaneously, dual pixel CMOS AF, and depth from defocus (DFD) AF method, have improved to the point that this deficiency is no longer an issue when compared to DSLR [6]. At the same time, developments in the area of image sensors, progressing from the M4/3

Appl. Sci. 2020, 10, 179; doi:10.3390/app10010179 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 179 2 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 15 progressing(micro four thirds)from the sensor M4/3 to (micro the full-frame four thirds) sensor, sensor have to been the full-frame steadily advancing. sensor, have In pursuitbeen steadily of the advancing.high-quality In achieved pursuit of by the the high-quality full-frame DSLR achieved in CSC, by the a CSC full-frame that adopted DSLR in an CSC, APS-C a CSC (advanced that adopted photo ansystem, APS-C type-C) (advanced sensor photo was system, introduced type-C) for thesensor first wa times introduced in Korea in for 2010 the andfirst atime CSC in that Korea adopted in 2010 a andfull-frame a CSC sensorthat adopted was released a full-frame in Japan sensor in 2013 was [ 7released,8]. in Japan in 2013 [7,8]. Meanwhile, the the digital camera industry in in 2017 was equipped with a 4K UHD (ultra-high definition)definition) 60 frame/s frame/s video recording function [[9].9]. The movie recording function began to be used when thethe cameracamera sensorsensor was was replaced replaced with with a digitala digital sensor. sensor. However, However, problems problems such such as AF as speedAF speed and AFand noise,AF noise, which werewhich detrimental were detrimental in video recording,in video recording, were gradually were solved gradually through solved technological through technologicaladvancements. advancements. Specifically, by Specifically, integrating by phase inte detectiongrating phase AF and detection contrast AF AF, and as describedcontrast AF, above, as describedto compensate above, for to thecompensate respective for AF the methods’ respective shortcomings, AF methods’ theshortcomings, speed and the accuracy speed ofand AF accuracy in CSC wereof AF improved.in CSC were The improved. phase detection The phase AF method’sdetection drawbackAF method’s is that drawback the actual is that focal the position actual focal does positionnot match does due not to match AF computation due to AF computation movement [move10]. Therefore,ment [10]. Therefore, if it is supplemented if it is supplemented with contrast with AF,contrast that focusesAF, that accurately, focuses accurate it is possiblely, it tois focuspossible on fast-movingto focus on objectsfast-moving even inobjects dark surroundings.even in dark surroundings.In addition, the In compensation addition, the methodcompensation with phase method difference with phase AF or difference DFD AF was AF usedor DFD to determineAF was used the tofocusing determine direction. the focusing Contrast direction. AF, that canContrast focus precisely,AF, that iscan essential focus precisely, for any AF is method.essential Additionally, for any AF method.a driving Additionally, source capable a ofdriving moving source the AF capable group of constantly moving andthe quicklyAF group is required.constantly For and this quickly purpose, is required.a stepper For motor this (STM), purpose, that a hasstepper a higher motor precision, (STM), that is used has asa higher a driving precision, source foris used moving as a thedriving lens. sourceHowever, for STMmoving is unable the lens. to generateHowever, fast STM AF is when unable the to focusing generate group fast AF is heavywhen the because focusing the torquegroup is heavynot large because [11]. Hence, the torque when is the not STM large is used[11]. asHence, a driving when source, the STM the focusingis used as group a driving must be source, lightened the focusingfor optical group system must design. be lightened In this study, for optical the AF system group designeddesign. In a this very study, wide anglethe AF optical group system designed with a verya lightweight wide angle internal optical focus system system. with The a lightweight inner focus internal system focus is a method system. in The which inner the focus inner system lens group is a methodmoves and in which focuses the while inner thelens first group and moves last groups and focu ofses the while optical the system first and are last stationary. groups of The the floating optical system isare a stationary. method in whichThe floating two or system more arbitrary is a method lens in groups which move two or and more focus. arbitrary The 24 lens mm groups field of moveview isand also focus. included The in24 the mm product field of range view of is major also cameraincluded manufacturers, in the product with range field of angles major typically camera manufacturers,used for outdoor with scenes field or angles celestial typically objects. used However, for outdoor for such scenes an angleor celestial of view, objects. the optical However, system for withsuch anan innerangle focusing of view, CSC the interchangeable optical system lenswith or an a floatinginner focusing interchangeable CSC interchangeable lens using APS-C lens sensor or a floatingor less, andinterchangea a full-frameble lens CSC using interchangeable APS-C sensor lens or capableless, and of a focusingfull-frame only CSC with interchangeable one lens has lens not capableyet been of developed. focusing only Therefore, with one in this lens study, has wenot discussyet been the developed. optical design Therefore, method in of this the study, very wide we discussangle optical the optical system design of an method internal of focusing the very type wide that angle can focusoptical with system only of one an lens internal in the focusing CSC using type a thatfull-frame can focus sensor. with Theonly objective one lens ofin designingthe CSC using and a building full-frame the sensor. system The is forobjective use in of an designing optical-based and buildingcamera system. the system Representative is for use in examples an optical-based of inner focuscamera system system. and Representa floating systemtive examples in camera of opticsinner focusare shown system in and Figure floating1. system in camera optics are shown in Figure 1.

(a) (b)

Figure 1. Layout of (a) inner focus system [12] and (b) ﬂoating system [13]. Figure 1. Layout of (a) inner focus system [12] and (b) floating system [13]. 2. Theory 2. Theory A First Order Fit Using Gaussian Brackets A First Order Fit Using Gaussian Brackets As paraxial ray tracing resembles the optical system, the characteristics of the optical system can be calculatedAs paraxial easily ray andtracing quickly resembles [14]. Inthe this optical case, syst theem, eﬀ ectivethe characteristics focal length of (EFL), the optical back focal system length can be(BFL), calculated and front easily focal and length quickly (FFL) [14]. of theIn this optical case, system the effective can be expressedfocal length by (EFL), Equations back (1)–(3)focal length using (BFL),a Gaussian and front bracket focal [15 length]. Here, (FFL)k represents of the optical the refracting system can power be expressed of the lens by surface, Equationsd represents (1)–(3) using the adistance Gaussian between bracket the [15]. front Here, and k represents rear surfaces theof refracting the lens, poweri represents of the thelensrefractive surface, d index,represents and the distance between the front and rear surfaces of the lens, i represents the refractive index, and the Appl. Sci. 2020, 10, 179 3 of 15 subscript is a number corresponding to each side of the optical system. From Equations (1)–(3), it is evident that the three variables must be adjusted to satisfy the desired EFL, BFL, and FFL [16]. " # p1 p2 pi 1 k1, k2, , , ki 1, − , ki = EFL, (1) − i1 − i2 ··· − − ii 1 − " # p1 p2 pi 1 k1, k2, , , ki 1, − = BFL, (2) − i1 − i2 ··· − − ii 1 − " # p1 p2 pi 1 k2, , , ki 1, − , ki = FFL. (3) − i1 − i2 ··· − − ii 1 − For example, the focal length EFL of an optical system consisting of n elements is expressed by the Gaussian bracket method and the third, sixth, and tenth elements, α3, α6, and α10, respectively, are extracted as indicated in Equation (4).

EFL = α1aiajak + α2aiaj + α3ajak + α4aiak + α5ai + α6aj + α7ak + α8, (4)

BFL = β1aiajak + β2aiaj + β3ajak + β4aiak + β5ai + β6aj + β7ak + β8, (5)

FFL = γ1aiajak + γ2aiaj + γ3ajak + γ4aiak + γ5ai + γ6aj + γ7ak + γ8, (6) 4 3 2 S1ak + S2ak + S3ak + S4ak + S5 = 0. (7) Here, element a represents the selected variable, refractivity or thickness, and the subscripts i, j, k represent the first, second, and third variables in order, and α, including the subscript, is defined to simplify the expansion of the Gaussian parenthesis [15]. By defining β and γ in the same way, BFL and FFL can be expressed simply as Equations (5) and (6). In order to obtain each variable, Equations (4)–(6) are combined and expressed as Equation (7), where S , S , S denote the coefficients of the 1 2 ··· 5 quadratic equation. Therefore, by obtaining the solution of Equation (7), it is possible to obtain design variables αi, αj, and αk satisfying an arbitrary focal length, back focal distance, and front focal length.

3. Methods

3.1. Primary Design Approximation of Optical Lens The design software used in this study was CODE V (Synopsis, Optical system design Inc., East Boothbay, ME, USA). Figure2 illustrates the optical path for the optical system of Example 1 from the patent JP20160012034A2P. In Figure2, beginning at the object side to the sixth surface, the object serves as a converter to rapidly change the angle of view, and the sixth surface to the twenty-seventh surface forms shape of a double-gauss type for the incoming rays. This optical system has optical specifications of F/# 1.4 and EFL of 24 mm, and the applied imaging sensor size satisfies 21.7 mm of full-frame. However, the AF system is a floating type optical system focusing on the third group from the 12th surface of the optical system to the 16th surface and the fourth and fifth groups from the 18th surface to the 27th surface. As it has a flange back distance for DSLR, it does not match the flange back distance for the CSC design. Appl. Sci. 2020, 10, 179 4 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 15

1 31 11 28 12 27 17

FigureFigure 2. 2.Optical Optical layout layout for for JP20160012034A2P, JP20160012034A2P, ExampleExample 1.

FigureFigure3 shows 3 shows the the optical optical path path for for the the optical optical system system of of ExampleExample 11 fromfrom thethe patentpatent KR1020140124286AP.KR1020140124286AP. The The first first to to sixth sixth surfaces surfaces on on the the object object sideside actact asas converters, and and the the seventh seventh surfacesurface to theto the 21st 21st surface surface serve serve as as the the imaging imaging unit, unit, and andthe the focusingfocusing is performed with with the the single single lenslens of the of the 13th 13th surface. surface. This This optical optical system system has has the the flange flange backback distancedistance for the the CSC CSC optical optical system system capablecapable of AF of AF with with internal internal focusing focusing using using a a single single F F/#/# of of 1.4, 1.4, andand thethe imaging sensor sensor has has aa full-frame full-frame sizesize of 21.7 of 21.7 mm mm or APS-Cor APS-C size size of of 14.25 14.25 mm mm is is not not satisfied. satisfied. Therefore,Therefore, by combining the the two two optical optical systemssystems appropriately, appropriately, basic basic design design data data that satisfiesthat satisfies both theboth F /#the of F/# 1.4, of AF 1.4, with AF the with internal the internal focusing methodfocusing using method a single using lens a single anda lens full-frame and a full-frame imaging imaging sensor sensor was obtained. was obtained. The 1stThe surface 1st surface to Theto The 11th surface of the Example No. 1 optical system from the patent JP 20160012034A2P (Figure 2) 11th surface of the Example No. 1 optical system from the patent JP 20160012034A2P (Figure2) and the 7th surface to The 21th surface of the Example No. 1 optical system from the patent No. and the 7th surface to The 21th surface of the Example No. 1 optical system from the patent No. KR1020140124286AP (Figure 3) were synthesized. Prior to combining the two optical systems, the KR1020140124286AP (Figure3) were synthesized. Prior to combining the two optical systems, the KR1020140124286AP optical system was adapted to meet the specifications of the APS-C sensor such KR1020140124286AP optical system was adapted to meet the specifications of the APS-C sensor such that it was extended to the size of a full-frame sensor. Each focal length of the synthesized lenses was thatcompared it was extended with the to optical the size systems of a full-frame of JP20160012034A2P sensor. Each focaland KR1020140124286AP, length of the synthesized and the lenses group was comparedwith minimum with the variance optical was systems then ofselected. JP20160012034A2P For example, the and focal KR1020140124286AP, distance from the 12th and surface the group to withthe minimum 27th surface variance of the wasJP20160012034A2P then selected. optical For example, system is the 43.508 focal mm, distance and the from focal the length 12th surfacefrom the to the7th 27th surface surface to of the the 21st JP20160012034A2P surface of the KR1020140124286AP optical system is 43.508 optical mm, system and is the 45.926 focal mm, length which from is thea 7threlatively surface to small the 21stfocal surface length difference of the KR1020140124286AP from other lens groups. optical The system objective is 45.926of the synthesis mm, which of the is a relativelytwo optical small systems focal length is to replace difference the imaging from other grou lensp of the groups. JP20160012034A2P The objective optical of the system synthesis with of the the twoimaging optical systemsgroup of isthe to KR1020140124286AP replace the imaging optical group system of the JP20160012034A2Pin such a way that the optical paraxial system optical with theproperties imaging group of the of optical the KR1020140124286AP system after synthesis optical would system not differ in such greatly a way from that the the paraxial paraxial optical optical propertiesproperties of theof the optical JP20160012034A2P system after optical synthesis system would befo notre synthesis. differ greatly Therefore, from the the optical paraxial system optical of propertiesJP20160012034A2P of the JP20160012034A2P is considered as optical an optical system system before having synthesis. a first group Therefore, from the the 1st optical surface system to the of JP20160012034A2P11th surface and is a consideredsecond group as anfrom optical the 12th system surface having to the a first27th group surface. from The the principal 1st surface points to of the 11theach surface group and are a determined second group and fromthe principal the 12th point surface of the to optical the 27th system surface. is −98.353 The principalmm from pointsthe 11th of eachsurface. group Additionally, are determined the and first theprincipal principal point point of the of second the optical group system is 39.255 is mm98.353 from mm the from12th surface the 11th − surface.of the Additionally, optical system, the the first optical principal distance point difference of the second between group the two is 39.255 groups mm is 142.852 from the mm 12th from surface the second principal point of the first group to the first principal point of the second group. Likewise, in of the optical system, the optical distance difference between the two groups is 142.852 mm from the the KR1020140124286AP optical system, if the 1st to 6th surfaces are referred to as the 1st group and second principal point of the first group to the first principal point of the second group. Likewise, the 7th surface to the 21st surface as the second group. The first principal point of the second group in the KR1020140124286AP optical system, if the 1st to 6th surfaces are referred to as the 1st group in the KR1020140124286AP optical system is separated by 27.772 mm from the 7th surface. As the and the 7th surface to the 21st surface as the second group. The first principal point of the second second group of the JP20160012034A2P optical system is replaced by the second group of the groupKR2020140124286AP in the KR1020140124286AP optical system, optical the distance system isfrom separated the second by 27.772principal mm point from of the first 7th surface.group As theof JP20160012034A2P second group of the to JP20160012034A2Pthe first principal poin opticalt of the system second is replacedgroup of byKR1020140124286AP the second group ofwas the KR2020140124286APsupposed to be identical optical at 142.852 system, mm. the Therefore, distance from the distance the second between principal the lenses point on of the the 11th first surface group of JP20160012034A2Pand the 12th surface toafter the synthesis first principal should pointbe increased of the by second 16.715 group mm, which of KR1020140124286AP is the distance between was supposed to be identical at 142.852 mm. Therefore, the distance between the lenses on the 11th surface Appl. Sci. 2020, 10, 179 5 of 15

Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 15 and the 12th surface after synthesis should be increased by 16.715 mm, which is the distance between thethe first first principal principal point point of of the the second second group group of of JP20160012034A2P JP20160012034A2P and the the first first principal principal point point of of the the 2nd2nd group group of KR1020140124286AP.of KR1020140124286AP. Specifications Specifications of the of opticalthe optical system system as shown as shown in Figure in Figure4 are shown4 are in Tableshown1. in Table 1.

7 6 24 12 21 22

FigureFigure 3. 3.Optical Optical layout layout forfor KR20140124286AP,KR20140124286AP, Example 1. 1.

TableTable 1. First 1. First orders orders of of optical optical layout layout after after combining combining thethe opticaloptical layout of of JP20160012034A2P JP20160012034A2P and and the the opticaloptical layout layout of KR1020140124286AP.of KR1020140124286AP.

EFLEFL 26.302 26.302 mm mm BFLBFL 10.830 10.830 mm mm FFLFFL 19.457 19.457 mm mm F/#F/# 1.4 1.4 Image distance 10.830 mm Image distance 10.830 mm Overall length 122.481 mm ParaxialOverall image length height 122.481 21.7 mm mm Paraxial image height 21.7 mm HFOV (Half Field of View) 39.5◦ HFOV (Half Field of View) 39.5°

TheThe optical optical path path of the of synthesized the synthesized optical optical system system is shown is inshown Figure in4 .Figure The EFL 4. ofThe the EFL synthesized of the opticalsynthesized system isoptical 26.302 system mm, which is 26.302 is the mm, di ffwhicherence is between the difference the designed between target the designed of 24 mm target and 2.3of 24 mm at 5mm degrees and 2.3 field mm of at view. 5 degrees In addition, field of theview. BFL In isaddition, 10.83 mm, the whichBFL is is10.83 shorter mm, thanwhich the is standardshorter than CSC flangethe standard back specification. CSC flange Considering back specification. that this Considering optical design that this is aoptical verywide design angle is a very system wide intended angle forsystem the CSC intended format, for proper the CSC correction format, of proper the focal correction length andof the back focal focal length length and is back essential focal andlength so thisis wasessential adjusted and by so solving this was the adjusted simultaneous by solving equations the simultaneous of Equation (2).equations Here, theof Equation lens surface (2). Here, selected the to correctlens eachsurface day’s selected radiation to correct must beeach capable day’s ofradiation physical must fabrication, be capable where of physical the curvature fabrication, and thickness where afterthe correction curvature mustand thickness not be largely after correction changed must from no thet be existing largely values. changed Additionally, from the existing as there values. is no realAdditionally, numeric solution as there in is the no real quadratic numeric equation solution toin the find quadratic the solution equation of the to find focal the distance, solution variousof the combinationsfocal distance, of variablesvarious combinations were applied, of andvariables then, were R1, R4, applied, and D11 and variables then, R1, wereR4, and selected. D11 variables R1 is the curvaturewere selected. of the firstR1 is surface, the curvature R4 is the of the curvature first surfac of thee, R4 fourth is the surface, curvature and of D11 the fourth is the lenssurface, thickness and of theD11 11th is the surface. lens thickness When theof the variables 11th surface. R1, R4, When and D11the variables were selected R1, R4, and and the D11 BFL were was selected corrected andto 15 mm,the BFL the was changed corrected values to 15 of mm, each the variable changed were values as illustrated of each variable in Table were2. Atas thisillustrated point, in if aTable second 2. At this point, if a second correction is performed to adjust the focal length to 24, it was confirmed that correction is performed to adjust the focal length to 24, it was confirmed that the parameters of R2, the parameters of R2, R3, and R25 variables changed as illustrated in Table 3. In this case, variable R3, and R25 variables changed as illustrated in Table3. In this case, variable selection was chosen by selection was chosen by examining the variable combinations as mentioned above. examining the variable combinations as mentioned above. Appl. Sci. 2020, 10, 179 6 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 15

FigureFigure 4. 4.Optical Optical layout layout after after combining combining the the optical optical layout layout of JP20160012034A2Pof JP20160012034A2P and and the opticalthe optical layout oflayout KR1020140124286AP. of KR1020140124286AP.

Table 2. Combined optical layout BFL and EFL change value before and after correction. Table 2. Combined optical layout BFL and EFL change value before and after correction. Before R1 53.9982 R4 28.8431 D11 16.7154 BFL Before R1 53.9982 R4 28.8431 D11 16.7154 BFL After 37.235853 18.078948 13.959187 After 37.235853 18.078948 13.959187 Before R2 25.7944 R3 45.1546 R25 −88.95502 EFL Before R2 25.7944 R3 45.1546 R25 88.95502 EFL − AfterAfter 20.84119820.841198 30.19839630.198396 −134.9441134.9441 − Figure 5 illustrates the optical path when the BFL of the synthesized optical system is adjusted andTable the 3.EFLSpot is sizecorrected and the to spherical 24 mm. aberration It is evident change that due the to light the radiation aberration correction is very of large the synthesized through the opticaloptical path system. of Figure 5. In the optical design software, design optimization is the process to minimize the size of the surface area. Allowing large aberrations in the central part of the basic design process RMS (mm) 100% (mm) SAA (mm) SAT (mm) can be detrimental in the early stages of the optimization process, especially as the center size is more (a) 3.571648 11.712321 0.873108 2.69256 sensitive to the surface size. Hence, design optimization− should proceed− after correcting the (b) 2.343004 9.375108 0 1.155744 aberration to some extent. To that end, the aspherical contribution to the Seidel− 3rd order aberration (c) 1.485043 7.553029 0.64045 0.000062 − was used [17]. As(d) the aim 1.029953of the aberration 6.184465 correction in the 1.1 basic design 0.843992 process is to further smoothen the design optimization at the initial stage, spherical aberration, the factor with a considerable effect on gathering the ray to one point with respect to the central part of the upper surface,Figure was5 illustrates corrected. the The optical equation path for when the aspher the BFLical of surface the synthesized is shown opticalin Equation system (8), is where adjusted c andrepresents the EFL the is correctedcurvature toand 24 y mm.represents It is evidentthe vertical that distance the light from aberration the optical is axis very when large the through ray falls the opticalon the path refracting of Figure surface.5. In A the4 and optical A6 are design the fourth- software, and sixth-order design optimization aspherical iscoefficients, the process respectively, to minimize theand size the of change the surface of the area. Seidel Allowing aberration large coefficient aberrations by the in thefourth-order central part aspherical of the basic coefficient design can process be canexpressed be detrimental as Equations in the (9)–(12) early stages [17]. of the optimization process, especially as the center size is more sensitive to the surface size. Hence, design2 optimization should proceed after correcting the aberration = cy + 4 + 6 to some extent. To that end,Sag theAS aspherical contributionA4 y to theA6 Seidely  3rd order aberration was used [17]. 1+ 1− c 2 y 2 As the aim of the aberration correction in the basic design process is to further, smoothen the design(8) 2 optimization at the initial stage, sphericaly 2 ()1 aberration,+ 8r 3 A y 4 the() factor1+16r with5 A ay considerable6 effect on gathering = + 4 + 6  the ray to one point with respect to2 ther central8r part3 of the upper16r surface,5 was corrected. The equation for the aspherical surface is shown in Equation (8), where c represents the curvature and y represents S = S + 8()n − n' A y4 the vertical distance from the optical axisI total whenI the ray falls4 on, the refracting surface. A4 and (9)A6 are S = S + 8()n − n' A y 3 y II total II 4 , (10) Appl. Sci. 2020, 10, 179 7 of 15 the fourth- and sixth-order aspherical coefficients, respectively, and the change of the Seidel aberration coefficient by the fourth-order aspherical coefficient can be expressed as Equations (9)–(12) [17].

2 cy 4 6 SagAS = + A4 y + A6 y 1+ √1 c2 y2 ··· − 2 , (8) 2 (1+8r3A )y4 (1+16r5A ) y6 = y + 4 + 6 2r 8r3 16r5 ··· 4 Appl. Sci. 2020, 10, x FOR PEER REVIEW S = SI + 8(n n0)A y ,7 of (9)15 I total − 4 3 SIItotal = SII + 8(n n0)A4 y y, (10) S = S + 8()n −−n' A y 2 y 2 III total III 4 2 2 , (11) S = SIII + 8(n n0)A y y , (11) III total − 4 S = S + 8()n − n' A yy 3 SV total = SV V + 8(n n0)A4 yy ,(12) (12) V total − 4 , wherewhere SSI Irepresentsrepresents the the coefficient coefficient of ofthe the spherical spherical aberra aberrationtion of the of theSeidel Seidel aberration aberration coefficient, coefficient, the subscriptthe subscript total totalrepresents represents the theSeidel Seidel aberration aberration coefficient coefficient including including the the aspherical aspherical surface, surface, nn representsrepresents the the refractive refractive index index immediately immediately before before the the refracting refracting surface, surface, nn’ ’represents represents the the refractive refractive indexindex of of the the refracting refracting surface, surface, yy isis the the height height of of the axial ray of the refractingrefracting surface,surface, SSIIII, SIII,, and and SSVV areare the the coma coma aberration, aberration, astigmatism, astigmatism, and and distortion distortion aberration, aberration, and and ȳy¯ representsrepresents the the principal principal ray ray heightheight at at the the corresponding corresponding refracting refracting surface. surface.

1 12 11 26 7 1617 21 6 18 19

FigureFigure 5. 5. OpticalOptical layout layout of of the the optical optical sy systemstem with with 24 24 mm mm EFL EFL increment. increment.

Moreover,Moreover, as the asphericalaspherical surfacesurface becomes becomes larger, larger, the the material material and and processing processing cost cost increase increase [18]. [18].Therefore, Therefore, the the 18th 18th surface surface was wa selecteds selected as as the the aspherical aspherical surface surface toto correctcorrect spherical aberration aberration consideringconsidering both both the the correction correction effect effect and and the the processi processingng cost. cost. In this In thisstudy, study, a method a method to reduce to reduce the totalthe totalnumber number of aberrations of aberrations by adjusting by adjusting the bala thence balance with higher with higherorder aberrations order aberrations was used. was Using used. Using Equation (9), the fourth-order aspheric coefficient A when the Seidel spherical aberration is Equation (9), the fourth-order aspheric coefficient A4 when the4 Seidel spherical aberration is zero was 5 zero was calculated as −51.982156e , and Figure6b shows the order di fference when it is applied to calculated as −1.982156e− , and Figure− 6b shows the order difference when it is applied to the 18th surface.the 18th In surface. this case, In by this comparing case, by comparing with Figure with 6a, which Figure is6a, the which order is difference the order before di fference the correction, before the itcorrection, can be confirmed it can be confirmed that the thatlongitudinal the longitudinal spherical spherical aberration aberration shifted shifted in the in the positive positive direction. direction. Additionally,Additionally, as as shown shown in in Table Table 33 line (b), SATT andand spot spot size size decreased decreased from from 3.571648 3.571648 to to 2.343004 2.343004 mm. mm. Here,Here, SA SATT isis the the sum sum of of the the third-order, third-order, fifth-orde fifth-order,r, and and seventh-order seventh-order sp sphericalherical aberrations. BasedBased on EquationsEquations (9)–(12),(9)–(12), when when only only the the spherical spherical aberration aberration is corrected, is corrected, the aspherical the aspherical surface surfaceis used is as used the surfaceas the surface closest clos to theest diaphragm, to the diaphragm, so that so the that principal the principal ray height ray isheight minimized, is minimized, thereby therebyreducing reducing the variation the variation of the coma of the aberration, coma aberra astigmatism,tion, astigmatism, and distortion and distortion aberration, aberration, and effectively and effectivelycorrecting correcting the spherical the aberration.spherical aberration. FigureFigure 6c6c shows the order of magnitude of the spherical aberration of the synthesized optical 5 system after secondary correction. In this case, A is calculated as 1.648908e −,5 and SA is increased to system after secondary correction. In this case, A4 4 is calculated as− −1.648908e− , and SAAA is increased to 0.64045 mm as shown in Table 3 line (c), but SAT is decreased to −0.00006 mm. In addition, the ordinal number difference in Figure 6d shows the difference in the number after the third correction of the synthesized optical system, where A4 is −2.1495534e−5. As shown in Table 3 line (d), while the SAA is 1.1 mm and SAT is not identical to the secondary correction, the spot size of the on-axis is the smallest. The third order spherical aberration of the synthesized optical system was corrected and the basic design data of the super focus and large diameter optical system to be designed in this study were obtained. Specifications of the optical system as shown in Figure 7 are shown in Table 4. Figure 7 shows the data for radiation and optical path of the basic design. Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 15

Appl. Sci. 2020, 10, 179 8 of 15

0.64045 mm as shown in Table3 line (c), but SA is decreased to 0.00006 mm. In addition, the ordinal T − number difference in Figure6d shows the di fference in the number after the third correction of the synthesized optical system, where A is 2.1495534e 5. As shown in Table3 line (d), while the SA is 4 − − A 1.1 mm and SA is not identical to the secondary correction, the spot size of the on-axis is the smallest. FigureT 6. Degree change variances of the aspheric coefficient of the combined optical system. The third order spherical aberration of the synthesized optical system was corrected and the basic designTable data of3. theSpot super size and focus the and spherical large diameteraberration opticalchange due system to the to beradiation designed correction in this of study the were obtained.synthesized Specifications optical system. of the optical system as shown in Figure7 are shown in Table4. Figure7 showsAppl. theSci. 2020 data, 10 for, x FOR radiation PEER REVIEW and optical path of the basic design. 8 of 15 RMS (mm) 100% (mm) SAA (mm) SAT (mm) (a) 3.571648 11.712321 −0.873108 −2.69256 (b) 2.343004 9.375108 0 −1.155744 (c) 1.485043 7.553029 0.64045 −0.000062 (d) 1.029953 6.184465 1.1 0.843992

Table 4. First orders of basic design.

EFL 24 mm BFL 15 mm FFL 19.457 mm F/# 1.488 Image distance 15 mm Overall length 119.724 mm Paraxial image height 21.7 mm HFOV 42.1°

Figure 6. Degree change variances of the aspheric coeﬃcient of the combined optical system. Figure 6. Degree change variances of the aspheric coefficient of the combined optical system. 1 12 11 Table 3. Spot size and the7 spherical aberration change due16 17 to the radiation21 correction of the 6 synthesized optical system. 18 19

RMS (mm) 100% (mm) SAA (mm) SAT (mm) (a) 3.571648 11.712321 −0.873108 −2.69256 (b) 2.343004 9.375108 0 −1.155744 (c) 1.485043 7.553029 0.64045 −0.000062 (d) 1.029953 6.184465 1.1 0.843992

Table 4. First orders of basic design.

EFL 24 mm BFL 15 mm Figure 7. Optical path diagram of basic design data. Figure 7. OpticalFFL path diagram of 19.457basic design mm data. Table 4. FirstF/# orders of basic1.488 design. Image distance 15 mm EFL 24 mm OverallBFL length 119.724 15 mm mm ParaxialFFL image height 21.7 19.457 mm mm HFOVF/# 42.1° 1.488 Image distance 15 mm Overall length 119.724 mm 1 Paraxial image height12 21.7 mm 11HFOV 42.1◦ 7 1617 21 6 18 19 3.2. Design Optimization and Evaluation Optimization must be performed to achieve the desirable resolution from the primary design approximation that was previously obtained [19,20]. The optical design software (CodeV, Synopsys,

Figure 7. Optical path diagram of basic design data. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 15

3.2. Design Optimization and Evaluation Appl. Sci. 2020, 10, 179 9 of 15 Optimization must be performed to achieve the desirable resolution from the primary design approximation that was previously obtained [19,20]. The optical design software (CodeV, Synopsys, MountainMountain View, View, CA, CA, USA), USA), is is used used forfor thethe optimization.optimization. During During the the optimization, optimization, infinite infinite object object distance, object magnification of 1/40 (approximately 1.1 m from the image sensor) for the principal distance, object magnification of 1/40×× (approximately 1.1 m from the image sensor) for the principal shootingshooting distance, distance, and and the the greatest greatest distance distance to to the the object object are are decided, decided, atat aa levellevel comparablecomparable to those of theof competitors’ the competitors’ products. products. Then, Then, the the optimization optimization was was implementedimplemented to to realize realize the the performance performance for for thethe object object at at a distancea distance of of approximately approximately 0.25 0.25 m m from from the the image image sensor. sensor. Moreover,Moreover, thethe designdesign goals of F#of/1.4 F#/1.4 large large aperture, aperture, the the very very wide wide angle angle of of 42 42◦ °HFOV HFOV (Half(Half Field of of View), View), and and inner inner focusing focusing of of 1-piece1-piece lens lens were were maintained. maintained. ForFor mechanicalmechanical constr constraints,aints, the the distance distance from from the the last last surface surface of ofthe the systemsystem to to the the image image surface surface was controlled controlled to to be be 16 16 mm mm or orlonger longer based based on its on availability its availability in CSC. in CSC.In Inaddition, addition, the the diameter diameter of of the the last last lens lens in in the the syst systemem was was limited limited to toavoid avoid vignetting vignetting of ofthe the principal principal rayray due due to to the the camera camera mount. moun Fort. For the the relative relative illumination, illumination, CRA CRA (Chief (Chief Ray Ray Angle) Angle) was was also also set set to to 21 ◦ or21° less or to less obtain to obtain 20% or20% higher or higher at the at cornerthe corner of the of screen.the screen. TheThe diagram diagram of of the the final final design design afterafter thethe optimization is is shown shown in in Figure Figure 8.8 .The The dotted dotted line line for for thethe AF AF lens lens above above indicates indicates the the traveltravel rangerange of the AF AF lens lens as as the the object object travels travels from from infinity infinity to tothe the nearestnearest focusing focusing distance. distance. Information Information on on curvature, curvature, thickness, thickness, material,material, etc.etc. for this optical system is mentionedis mentioned in Korean in Korean Patent Patent 10-2018-0123849. 10-2018-0123849.

AF lens Asphere ED lens Position1 infinity

Position4 0.25m from image surface 0.17x (-1/5.8)

FigureFigure 8.8. Diagram of of the the final final design. design. 4. Results and Discussion 4. Results and Discussion 4.1. Lens Performance Verification 4.1. Lens Performance Verification Three lenses, one 10th AF lens, one second lens, and one 14th lens, were used as the aspheric Three lenses, one 10th AF lens, one second lens, and one 14th lens, were used as the aspheric surfacessurfaces in in the the design. design. TheThe totaltotal comprisedcomprised of of 14 14 lenses, lenses, including including two two extra extra low low dispersion dispersion (ED) (ED) lenses.lenses. The The aspherical aspherical surface surface of of thethe AFAF lenslens was us useded for for spherical spherical aberration aberration correction correction in inthe the basic basic designdesign process. process. Additionally, Additionally, toto correctcorrect residualresidual aberrations such such as as coma coma aberration, aberration, astigmatism, astigmatism, andand distortion distortion aberration, aberration, thethe objectobject side side and and the the uppe upperr lens lens side side was was used. used. Figure Figure 9 is 9a iscollection a collection of ofgraphs illustrating illustrating the the species species difference difference between between the the infinity infinity and and the the principal principal shooting shooting distance. distance. TheThe pitch pitch angle angle di differencefference is is generally generally stablestable atat infinityinfinity and and the the principal principal shooting shooting distance, distance, and and the the distortiondistortion aberration aberration is is −5%,5%, whichwhich includesincludes some barrel barrel distor distortiontion but but still still meets meets the the design design goal. goal. − FiguresFigures 10 10 and and 11 11 illustrate illustrate the the TV TV distortion distortion and and MTF MTF graph graph of of the the final final design. design. InIn infinity,infinity, the the TV distortionTV distortion has a has Barrel a Barr typeel type distortion distortion of 1.40%of −1.40% and and+1.57% +1.57% at theat the furthest furthest object object distance, distance, which which has − a pincushionhas a pincushion type of type distortion. of distorti Theon. MTF The MTF satisfies satisfies 70% or70% more or more at 30 at lp 30/mm lp/mm with with respect respect to the to centerthe ofcenter the screen, of the and screen, more and than more 30% than at the30% periphery at the periphery up to theup 0.85-field.to the 0.85-field. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 15 Appl. Sci. 2020, 10, 179 10 of 15 656.2725 NM 587.5618 NM Appl. Sci. 2020, 10, x FOR PEER REVIEW 546.0740 NM10 of 15 486.1327 NM 435.8343 NM

656.2725 NM 587.5618 NM LONGITUDINAL ASTIGMATIC 546.0740 NM DISTORTION486.1327 NM SPHERICAL ABER. FIELD CURVES 435.8343 NM IMG HT IMG HT LONGITUDINAL ASTIGMATICS T 1.00 21.70 DISTORTION21.70 SPHERICAL ABER. FIELD CURVES IMG HT IMG HT S T 1.00 21.70 21.70 0.75 16.28 16.28 Position 1 Infinity 0.75 16.28 16.28 Position 1 0.50 10.85 10.85 Infinity 0.50 10.85 10.85

0.25 5.43 5.43

-0.20 -0.10 0.0 0.10 0.20 -0.200 -0.100 0.0 0.100 0.200 -5.0 -2.5 0.0 2.5 5.0 FOCUS (MILLIMETERS) FOCUS (MILLIMETERS) % DISTORTION

-0.20 -0.10 0.0 0.10 0.20 -0.200 -0.100 0.0 0.100 0.200 -5.0 -2.5 0.0 2.5 5.0 FOCUS (MILLIMETERS) FOCUS (MILLIMETERS) % DISTORTION LONGITUDINAL ASTIGMATIC DISTORTION SPHERICAL ABER. FIELD CURVES LONGITUDINAL ASTIGMATIC IMG HT DISTORTIONIMG HT SPHERICAL ABER. T FIELD CURVESS 1.00 21.70 21.70 IMG HT IMG HT T S 1.00 21.70 21.70

0.75 16.28 16.28

Position 2 0.75 16.28 16.28

m=-1/40Position 2 0.50 10.85 10.85

m=-1/40 0.50 10.85 10.85

0.25 5.43 5.43

-0.20 -0.10 0.0 0.10 0.20 -0.20 -0.10 0.0 0.10 0.20 -5.0 -2.5 0.0 2.5 5.0 -0.20FOCUS -0.10 (MILLIMETERS) 0.0 0.10 0.20 -0.20FOCUS -0.10 (MILLIMETERS) 0.0 0.10 0.20 -5.0 -2.5% DISTORTION 0.0 2.5 5.0 FOCUS (MILLIMETERS) FOCUS (MILLIMETERS) % DISTORTION

Figure 9. Spherical aberration degree of the final design at the infinity and principal shooting distance. FigureFigure 9. 9.Spherical Spherical aberration aberration degree degree of of thethe ﬁnalfinal design at at the the infinity inﬁnity and and principal principal shooting shooting distance. distance.

Position 1, infinity, Barrel Type Position 4, m=-0.19, Pincushion Type Position 1, infinity, Barrel Type Position 4, m=-0.19, Pincushion Type 1.5738% -1.3973% 1.5738% -1.3973% VERTICAL FOV VERTICAL FOV VERTICAL FOV VERTICAL FOV

-5.0608%-5.0608% 3.1182%3.1182%

-0.3570%-0.3570% ParaxParax FOVFOV HORIZONTALHORIZONTAL FOV FOV 0.9954% HORIZONTALHORIZONTAL FOV FOV ActualActual FOVFOV Figure 10. FigureFigure 10. 10. TVTV TV distortion distortion distortion of ofof the thethe final ﬁnalfinal designs designs at the infiniteinfinite inﬁnite and and nearest nearest distance. distance. Appl. Sci. 2020, 10, 179 11 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 15

10 LP/MM (sagittal) 10 LP/MM (sagittal) infinity 10 LP/MM (tangential) -1/40 10 LP/MM (tangential) 30 LP/MM (sagittal) 30 LP/MM (sagittal) 10 LP/MM (sagittal) 10 LP/MM (sagittal) 30 LP/MM (tangential) 30 LP/MM (tangential) infinity 10 LP/MM (tangential) -1/40 10 LP/MM (tangential) MTF : 30 % line MTF : 30 % line 30 LP/MM (sagittal) 30 LP/MM (sagittal) 30 LP/MM (tangential) 30 LP/MM (tangential) 1.0 MTF : 30 % line 1.0 MTF : 30 % line

1.00.9 1.00.9

0.90.8 0.90.8

0.80.7 0.80.7

0.70.6 0.70.6

0.60.5 0.60.5

0.50.4 0.50.4 MTF(Diffraction) MTF(Diffraction)

0.40.3 0.40.3 MTF(Diffraction) MTF(Diffraction)

0.30.2 0.30.2

0.20.1 0.20.1

0.10.0 0.10.0 0.00 5.00 10.00 15.00 20.00 25.00 0.00 5.00 10.00 15.00 20.00 25.00 REAL IMAGE HEIGHT (mm) REAL IMAGE HEIGHT (mm) 0.0 0.0 0.00 5.00 10.00 15.00 20.00 25.00 0.00 5.00 10.00 15.00 20.00 25.00 REAL IMAGE HEIGHT (mm) REAL IMAGE HEIGHT (mm) FigureFigure 11. 11.Image Image heights heights vs. vs. MTF MTF plotsplots ofof thethe finalfinal design designss at at the the infinite infinite and and main main imaging imaging distance. distance. Figure 11. Image heights vs. MTF plots of the final designs at the infinite and main imaging distance. AsAs shown shown in Figurein Figure 11, as11, the as imagethe image height height increases, increases, the MTF the MTF gradually gradually decreases. decreases. This isThis because is aberrationsbecauseAs shownaberrations such asin comaFigure such andas11, coma as astigmatism the and image astigmatism increaseheight increases,increase as the imageas the the MTF image height gradually height increases, increases, decreases. as shown as Thisshown in is the astigmaticbecausein the astigmatic aberrations field curve field such in curve Figure as coma in 9Figure. and Most astigmatism9. cameraMost camera manufacturers increase manufactur as the disclose imageers disclose height MTFs MTFs increases, with with 10lp as10lp/mm /shownmm and 30inand lp /themm 30 astigmatic lp/mm spatial spatial frequencies field frequencies curve for in cameraFigure for camera 9. optics Most optics [ 21camera,22 [21,22].]. Therefore, manufactur Therefore, thisers paper thisdisclose paper also MTFs also identified identifiedwith 10lp/mm MTF MTF based onandbased the 30 same on lp/mm the criteria. same spatial criteria. frequencies for camera optics [21,22]. Therefore, this paper also identified MTF basedFigureFigure on the12 12 showssame shows criteria. the the MTF MTF as as the the spatialspatial frequencyfrequency ch changes.anges. Figure Figure 12a 12 ashows shows the the object object at atinfinity infinity Figure 12 shows the MTF as the spatial frequency changes. Figure 12a shows the object at infinity andand Figure Figure 12 b12b shows shows the the magnification magnification of of1/ 40−1/40 times. times. The The maximum maximum spatial spatial frequency frequency in each in each graph and Figure 12b shows the magnification −of −1/40 times. The maximum spatial frequency in each is 135graph lp/ mm,is 135 which lp/mm, is which the Nyquist is the Nyquist frequency frequency of the digital of the sensor digital with sensor the with largest the pixellargest of pixel the currently of the graphcurrently is 135 available lp/mm, camera which isbody. the Nyquist frequency of the digital sensor with the largest pixel of the available camera body. currently available camera body.

Figure 12. Frequency vs. MTF plots of the final designs at the infinite and main imaging distance. FigureFigure 12. 12.Frequency Frequency vs. vs. MTFMTF plotsplots of the final ﬁnal designs designs at at the the infinite inﬁnite and and main main imaging imaging distance. distance. Appl. Sci. 2020, 10, 179 12 of 15

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 15 4.2. Tolerance Analysis 4.2. Tolerance Analysis A camera optical system is a relatively large product. In optical systems such as the one devised A camera optical system is a relatively large product. In optical systems such as the one devised in this study, many lenses are used, and a driving device for moving the lenses is used too. Therefore, in this study, many lenses are used, and a driving device for moving the lenses is used too. Therefore, it isit expensiveis expensive to to manufacture manufacture such such an an optical optical system.system. Consequently, Consequently, additional additional costs costs are are incurred incurred fromfrom disposal, disposal, if if a failurea failure occurs. occurs. However, However, defectdefect rates will will exist exist when when the the lens lens is is assembled assembled at atthe the massmass production production stage stage and and tested tested for for resolution, resolution, such as MTF. MTF. The The fundamental fundamental cause cause of of such such defects defects is thatis that the the lenses lenses are are decentered decentered or or tilted tilted at at thethe assemblyassembly stage. Nevertheless, Nevertheless, resolution resolution performance performance cancan be be improved improved by by decentering decentering or or tilting tilting aa specificspecific lens group. Figure Figure 13 13 is is a ahistogram histogram showing showing the the sensitivitiessensitivities of of spherical spherical aberration aberration and and fieldfield curvaturecurvature based on on decentering decentering and and tilting tilting of ofeach each lens lens groupgroup constituting constituting the the designed designed optical optical system. system. InIn Figure Figure 13 13,, SAM SAM isis spherical aberration, aberration, SAG SAG is the is thesagittal sagittal component component of field of curvature, field curvature, and andTAN TAN is isthe the tangential tangential component component of offield field curvatur curvature.e. As As illustrated illustrated in inFigure Figure 13, 13 the, the tangential tangential componentcomponent of theof the field field curvature curvature changes changes considerably considerably with respectwith respect to tilt, andto tilt, the and spherical the spherical aberration changesaberration with changes respect with to the respect decentering to the decentering of the third of group. the third Therefore, group. Therefore, it is possible it is topossible correct to the sphericalcorrect aberrationthe spherical and aberration field curvature and field caused curvat byure the caused assembly by the error assembly of the designed error of the optical designed system. It isoptical also evident system. thatIt is thealso sagittal evident component that the sagitta of thel component field curvature of the doesfield notcurvature change does significantly not change due to thesignificantly assembly due error to the of the assembly lens group. error of This the can lens be group. explained This can by thebe explained third order by aberration the third order theory, becauseaberration the sagittal theory, component because the is generallysagittal component about one thirdis generally smaller about than theone tangential third smaller component than the [23 ]. tangential component [23]. Manufacturing errors for the optical system can be classified as being rotationally symmetrical Manufacturing errors for the optical system can be classified as being rotationally symmetrical and asymmetrical with respect to the optical axis. Here, rotationally symmetrical errors include lens and asymmetrical with respect to the optical axis. Here, rotationally symmetrical errors include lens shape, thickness, and material refractive index, and rotational asymmetry errors include centering, shape, thickness, and material refractive index, and rotational asymmetry errors include centering, seat tilt, clearance, group decenter, and group tilt [24]. These errors allow us to calculate spherical seat tilt, clearance, group decenter, and group tilt [24]. These errors allow us to calculate spherical aberrationsaberrations and and field field curvatures curvatures that that aaffectffect resolutionresolution degradation. In In this this case, case, the the aberration aberration may may bebe corrected corrected by by adjusting adjusting the the decenter decenter andand tilttilt ofof thethe specific specific lens lens group group in in order order to to counteract counteract the the aberrationsaberrations generated generated during during the the assembly assembly ofof thethe opticaloptical system.

FigureFigure 13. 13.The The sensitivities sensitivities forfor the decenter and and tilt tilt of of the the lens lens group. group.

FigureFigure 14 14 illustrates illustrates the the results results ofof sphericalspherical aberrationaberration and and field field curvature curvature distribution distribution of ofthe the wholewhole optical optical system system by by randomly randomly generatinggenerating the above-mentioned errors errors of of assembly assembly and and parts, parts, andand also also illustrates illustrates the the change changeof of aberrationaberration distributiondistribution based based on on the the decentering decentering or or tilting tilting of ofa a specificspecific lens lens group. group. The The blue blue dot representsdot represents the aberration the aberration calculated calculated by the randomby the random number number generated error.generated The red error. dot denotesThe red thatdot denotes the spherical that the aberration spherical isaberration corrected is by corrected decentering by decentering of the third of group. the Thethird green group. dot indicatesThe greenthat dot indicates the field curvaturethat the field is cu correctedrvature is by corrected tilting the by firsttilting group. the first The group. purple The box herepurple denotes box ahere performance denotes a performance limit, and a limit, resolution and a boxresolution within box this within range this indicates range indicates good resolution good Appl. Sci. 2020, 10, 179 13 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 15 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 15 performance. The rest of the spherical aberrations remain after group 1 adjustment, which in turn resolution performance. The rest of the spherical aberrations remain after group 1 adjustment, which mayin leadturn tomay defects. lead to Therefore, defects. Therefore, this should this be shou consideredld be considered in the optical in the design optical stage design so thatstage there so that is no sphericalthere is aberrationno spherical according aberration to according the tilt of to group the tilt 1. of group 1.

Figure 14. The correction for the designed optical system aberrations. FigureFigure 14. 14.The The correction correction forfor the designed optical optical system system aberrations. aberrations. To estimate the actual failure rate from the results shown in Figure 12, we need to analyze the ToTo estimate estimate the the actual actual failure failure rate rate fromfrom thethe resultsresults shown in in Figure Figure 12, 12 we, we need need to toanalyze analyze the the process capability index (Cpk). Cpk can be calculated by Equation (13) [25]. processprocess capability capability index index (C (Cpk).).C Cpk can be calculated calculated by by Equation Equation (13) (13) [25]. [25 ]. pk pk =−() CKCpk=−()1 p CKCpk1 p Cpk =USL(1 −K) LSLCp , (13) = USLUSL− −LSL LSL,, (13) (13) CCpp== − Cp 6σσ 6σ where USL represents the upper speciﬁcation limit, LSL represents the lower speciﬁcation limit, where USL represents the upper specification limit, LSL represents the lower specification limit, σ σ representswhere USL the represents standard the deviation upper specification of the distribution, limit, LSL and representsK represents the lower kurtosis. specification limit, σ represents the standard deviation of the distribution, and K represents kurtosis. representsHere, USL the andstandardLSL aredeviation values of determined the distribution, from and the K inspection represents standard kurtosis. of the optical system. Here, USL and LSL are values determined from the inspection standard of the optical system. Most interchangeableHere, USL and LSL lens are companies values determined are based from on thethe spatialinspection frequency standard 30 oflp the/mm optical of the system. optical Most interchangeable lens companies are based on the spatial frequency 30 lp/mm of the optical system,Most interchangeable and the corresponding lens companies pixel size are of based the image on the sensor spatial is aboutfrequency 16.7 30µm. lp/mm If the of depth the optical of focus system, and the corresponding pixel size of the image sensor is about 16.7 μm. If the depth of focus system, and the corresponding pixel size of the image sensor is about 16.7 μm. If the depthUSL of focusLSL is determinedis determined by by considering considering the the F F/#/# ofof thethe opticaloptical system, then then this this value value determines determines USL andand LSL. . Theis Cdeterminedof each distributionby considering for the Equation F/# of the (13) optical is shown system, in Figure then this 15. value determines USL and LSL. Thepk Cpk of each distribution for Equation (13) is shown in Figure 15. The Cpk of each distribution for Equation (13) is shown in Figure 15.

Figure 15. The process capability index for the designed and assembled optical system. FigureFigure 15. 15.The The process process capability capability indexindex for the de designedsigned and and assembled assembled optical optical system. system. Appl. Sci. 2020, 10, 179 14 of 15

5. Conclusions Contrast detection type AF is required to achieve high accuracy movement when focusing drive source. As the stepping motor used to satisfy this requirement does not have large torque, it is difficult to realize fast AF when the focusing lens is heavy. The contrast AF method is basically used in CSC so that AF group weight would be considered in optical design stage to realize fast AF speed. Hence, basic design was carried out for the optical system that moves only one sheet for focusing. At initial design, two optical systems that partially satisfy the design specifications were synthesized. A typical floating type optical system with a large aperture and a very wide angle and an optical system with a large aperture and focusing with a single lens were appropriately synthesized. After combining the two optical systems, optical system curvature and lens space were adjusted using Gaussian bracket method to match the EFL, BFL, FFL, and so on to design target values. The final design obtained through optimization resulted in the optical system with satisfactory MTF performance of more than 70% at the center of the screen and at least 30% below 0.85-field for a spatial frequency of 30 lp/mm. Through this research, the world’s first CSC adopting a full-frame sensor can reduce the weight of one focusing lens and design a large-aperture ultra-wide-angle optical system that focuses internally. The same can be applied to a wider variety of wide angle and large aperture optical systems with a similar method. Additionally, products with high MTF performance are expected to meet the demand for the interchangeable lens family for a full-frame CSC that can be applied to a high-performance camera.

Author Contributions: Conceptualization, J.R.; Methodology, H.C. and J.R.; Software, J.R.; Data curation, J.R.; Writing—Original Draft Preparation, H.C. and J.R. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2017R1D1A3B03029119). This work (Grants No. S2563336) was supported by Business for Cooperative R&D between Industry, Academy, and Research Institute funded Korea Small and Medium Business Administration in 2019. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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