Optical Design of a Novel Collimator System with a Variable Virtual-Object Distance for an Inspection Instrument of Mobile Phone Camera Optics

applied sciences

Article Optical Design of a Novel Collimator System with a Variable Virtual-Object Distance for an Inspection Instrument of Mobile Phone Camera Optics

Hojong Choi 1, Se-woon Choe 1,2,* and Jaemyung Ryu 3,*

1 Department of Medical IT Convergence Engineering, Kumoh National Institute of Technology, Gumi 39253, Korea; [email protected] 2 Department of IT Convergence Engineering, Kumoh National Institute of Technology, Gumi 39253, Korea 3 Department of Optical Engineering, Kumoh National Institute of Technology, Gumi 39253, Korea * Correspondence: [email protected] (S.-w.C.); [email protected] (J.R.); Tel.: +82-54-478-7781 (S.-w.C.); +82-54-478-7778 (J.R.)

Featured Application: Optical inspection instrument for a mobile phone camera.

Abstract: The resolution performance of mobile phone camera optics was previously checked only near an infinite point. However, near-field performance is required because of reduced camera pixel sizes. Traditional optics are measured using a resolution chart located at a hyperfocal distance, which can only measure the resolution at a specific distance but not at close distances. We designed a new collimator system that can change the virtual image of the resolution chart from infinity to a short distance. Hence, some lenses inside the collimator systems must be moved. Currently, if the focusing lens is moved, chromatic aberration and field curvature occur. Additional lenses are required to Citation: Choi, H.; Choe, S.-w.; Ryu, correct this problem. However, the added lens must not change the characteristics of the proposed J. Optical Design of a Novel collimator. Therefore, an equivalent-lens conversion method was designed to maintain the first-order Collimator System with a Variable and Seidel aberrations. The collimator system proposed in this study does not move or change the Virtual-Object Distance for an resolution chart. Inspection Instrument of Mobile Phone Camera Optics. Appl. Sci. 2021, Keywords: collimation system; convertor system; mobile optics; phone camera optics 11, 3350. https://doi.org/10.3390/ app11083350

Academic Editor: Ye Zhi Ting 1. Introduction

Received: 15 March 2021 Phone cameras released in the early 2000s had a very large pixel size because of the Accepted: 5 April 2021 low number of pixels [1]. For this reason, tolerance for the sensor assembly error was large, Published: 8 April 2021 and no serious issue occurs without needing to correct the change in the image surface according to the object distance. However, the phone camera installed on the body of Publisher’s Note: MDPI stays neutral the latest phones has a very high number of pixels. Therefore, correcting the change in with regard to jurisdictional claims in the image surface is necessary according to the distance of the object. This image-surface published maps and institutional afﬁl- correction is called a focus adjustment, which is a technique commonly used in compact iations. cameras and interchangeable lenses [2]. However, because the phone camera is very small and light, it needs to perform focusing by moving the entire optical system [3]. Several studies on such phone cameras have been conducted. After focusing on the optical system, a study was conducted on the equipment that automatically checks Copyright: © 2021 by the authors. the uniformity of the image and various defects in the optical system [3–6]. Mostly, the Licensee MDPI, Basel, Switzerland. captured images of compact camera modules assembled onto a charge-coupled device This article is an open access article or complementary metal–oxide–semiconductor package, were automatically checked to distributed under the terms and detect some errors due to human operators. The proposed inspection processes that use conditions of the Creative Commons a specialized image-processing algorithm were designed to reduce the overall assembly Attribution (CC BY) license (https:// time and defects for compact camera module assembly. In addition, a real-time non- creativecommons.org/licenses/by/ contact eccentric-error-measurement system that uses an optical setup without a rotating 4.0/).

Appl. Sci. 2021, 11, 3350. https://doi.org/10.3390/app11083350 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3350 2 of 11

and clamping mechanism [7] and a method for detecting foreign substances on the lens surface were also proposed [8,9]. The system could assess the 2D surface defects and non-uniformity of the 3D film thickness of specific materials using pattern recognition and morphological techniques. Because distinguishing the defect between the intensity of the defect itself and the background is difficult, an image-processing algorithm that uses multiple Laplacian of Gaussian masks were implemented to detect the defects of the micro lens. Moreover, a dynamic focus region from several narrow region of interests across the vision screen was used to increase the sharpness and determine the focal distance and focal point for better image acquisition [10]. Various computation methods that use the gray-level difference, Tenengrad measurement, correlation evaluation, statistical approaches, Fourier transform, wavelet, and edge-based measurement have helped to reduce the processing time and increase the sharpness [10]. The computation methods could be obtained from a flat surface at a single focal distance. Multi-focus application from the synthesis of differently focused images has been investigated to evaluate the sharpness. However, these studies did not focus on the measurement of the resolution of optical systems. In addition, a complex algorithm-based inspection procedure was automatically performed, but detecting the defects, error status, and accuracy increment would take time. In the present paper, we propose a method that can measure the resolution according to the change in the object distance without moving the resolution chart. One of the common methods of checking a phone camera is to take a target, such as an ISO 12,233 resolution chart, using an optical system for the testing, and to check the various images inside the chart [11]. At this time, the resolution can change according to the object distance of the tested optical system so that it can also be evaluated. To perform this process, the resolution chart should be placed on the actual object distance to be evaluated. Recently, phone cameras have been capable of auto-focusing. Thus, the resolution chart should be moved during the resolution measurement. In this case, the size of the resolution chart must be varied according to its location. Figure1 shows the inspection principle of mobile phone camera optics. Figure1 shows that the change in the size of the resolution chart can be easily recognized when the object distance changes. However, the size of the resolution chart printed on a specific size cannot be changed according to the object distance. In addition, the space for measuring the resolution of the test lens increases as the inspection distance and resolution chart size increase. Therefore, another method must be employed to solve this technical problem. To reduce this inspection space for mobile phone camera optics, we also need to reduce the inspection distance. For this technical issue, the optical system must be placed between the test lens and resolution chart. This optical system should be constructed so that the resolution chart is enlarged to the desired inspection distance of the test lens. Hence, we need to use a converter or collimator. A converter is an optical system that converts the field angle of the optical system, such as the camera lens [12,13]. Meanwhile, a collimator is an optical system that creates a collimated beam of light to diverge from an object at a finite distance [14,15]. In the present study, a small resolution chart located at a finite distance must create a virtual image that is greatly enlarged over a long distance so that the features of the converter and collimator are well utilized. In this study, the optical system needed to measure the test lens is called a collimator. A collimator that reduces the inspection space of mobile phone camera optics has recently been studied [16,17]. The collimator is an optical system that converts the beam emitted from the original point source into a collimated beam. As the distance from the collimator to the point source gets closer, the beam passing through the collimator becomes a diverged beam. At this time, the point where the extension line of the diverged beam meets the optical axis is the virtual image. Here, if the test lens is placed in front of the collimator (opposite the point source), the test lens can see the virtual image. Therefore, from the point of view of a test lens, it becomes a virtual object. Therefore, in the present work, we propose a new design method for a collimator that can measure resolution at various distances of the resolution chart. The virtual-image position of the resolution chart must be varied so that the test lens can Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 12

a test lens, it becomes a virtual object. Therefore, in the present work, we propose a new design method for a collimator that can measure resolution at various distances of the resolution chart. The virtual-image position of the resolution chart must be varied so that the test lens can be inspected at various distances. To implement this function, the virtual- image position must be varied by moving a specific lens inside the collimator. Therefore, a collimator with a focus-adjustable capability must be used. Section 2 describes the design process of our proposed focus-adjustable collimator Appl. Sci. 2021, 11, 3350 system and illustrates the result of the paraxial power arrangement of the system. Section3 of 11 3 presents the layout of an aberration-balanced collimator system at infinite and close po- sitions to verify the capability of the proposed system. In addition, the modulation transfer-function curve of the designed collimator system is shown to verify the desirable op- betical inspected system performance at various distances. in inspecting To implement the general this phone function, cameras the because virtual-image the modulation position musttransfer be variedfunction by (MTF) moving must a speciﬁc be evaluated lens inside to commercialize the collimator. Therefore,the opticala system collimator in the with in- adustry. focus-adjustable Section 4 presents capability the must conclusion be used. of this study.

Figure 1. Test layout of the mobile phone camera optics. Figure 1. Test layout of the mobile phone camera optics.

2. MaterialsSection2 and describes Methods the design process of our proposed focus-adjustable collimator systemFigure and 2 shows illustrates a layout the of result the paraxial of the optical paraxial path power of our arrangement proposed focus-adjustable of the system. Sectioncollimator.3 presents Here, k the is layoutthe refracting of an aberration-balanced power of each lens collimatorgroup, which system is reciprocal at inﬁnite to and the closefocal positionslength of toeach verify lens the group capability [18,19]. of z the is the proposed distance system. between In addition, the lens groups, the modulation and u is transfer-functionthe angle of the axial curve ray of [20]. the designedThe subscripts collimator of these system variables is shown indicate to verify the number the desirable of lens opticalgroup, systemand “0” performance (zero) indicates in inspecting the object the plane. general k1–k3 phone indicate cameras the refracting because power the modula- of the tionlens transfergroup that function constitutes (MTF) the must collimator be evaluated system, to commercializeand k is the refrac theting optical power system of the in theop- industry.tical system Section to be4 tested.presents the conclusion of this study.

2. Materials and Methods Figure2 shows a layout of the paraxial optical path of our proposed focus-adjustable collimator. Here, k is the refracting power of each lens group, which is reciprocal to the focal length of each lens group [18,19]. z is the distance between the lens groups, and u is the angle of the axial ray [20]. The subscripts of these variables indicate the number of lens group, and “0” (zero) indicates the object plane. k1–k3 indicate the refracting power of the lens group that constitutes the collimator system, and k is the refracting power of the optical system to be tested. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 12

The axial ray from the target passes through the collimator and enters the test lens as a collimated beam. At this time, when some lens groups in the collimator system are moved, a diverged beam (not a collimated beam) is incident to the test lens. Therefore, the point where the axial ray incident to the test lens meets the optical axis becomes the object viewed by the test lens. In Figure 2, l denotes the object distance from the virtual target to the test lens. Here, when some lens groups of the collimator system are moved, the object distance can be varied from infinity to object distance l. However, the target position is fixed, although the target size does not change. Δf in Figure 2 denotes the amount of movement of the lens group that moves in the collimator system, and Δl denotes the amount of change in the image plane of the test lens. In Figure 2, k represents the refracting power of each lens group. Thus, it is a value provided by the optical design of the collimator system. In addition, l can be considered Appl. Sci. 2021, 11, 3350 as the minimum inspection distance. When k and l are given, Δl represents the value 4that of 11 can be calculated from the imaging equation [21].

l k k1 k2 3 k

u u 3 u0 4 z z z z bf 0 1 2 3 f Δl

Δf virtual target target stop

FigureFigure 2. 2. ParaxialParaxial layout layout of of our our proposed proposed focus-adjustable focus-adjustable collimator. collimator.

BeforeThe axial the lens ray fromgroup the moves target in passes the collimat throughor optical the collimator system, andthe collimated enters the beam test lens is incidentas a collimated to the test beam. lens, At and this the time, magnification when some of lens the groupscollimator in the optical collimator system system becomes are infinite.moved, If a divergedthe magnification beam (not is a collimatedcalculated beam)using isthe incident Gaussian tothe bracket test lens. at the Therefore, refracting the point where the axial ray incident to the test lens meets the optical axis becomes the object power and interval of each lens group, the result is expressed in Equation (1) [22,23]. In viewed by the test lens. Equation (1), minf denotes the magnification when the collimated beam is incident to the In Figure2, l denotes the object distance from the virtual target to the test lens. Here, test lens, and this value is infinite. z and k are the same as the values shown in Figure 2. when some lens groups of the collimator system are moved, the object distance can be varied from infinity to object distance l. However, the1 target position is fixed, although the []−−−zk,, zk ,, zk , == 0 (1) target size does not change.01∆f in 12 Figure2 23 denotes the amount of movement of the lens minf group that moves in the collimator system, and ∆l denotes the amount of change in the imageIn addition, plane of the the test refractive lens. power KCL of the collimator system can be expressed by EquationIn Figure (2), using2, k representsthe Gaussian the bracket. refracting power of each lens group. Thus, it is a value [ −−] = provided by the optical designkzkzkK11223,,,, of the collimator system.CL In addition, l can be considered(2) as the minimum inspection distance. When k and l are given, ∆l represents the value that can be calculatedFigure 2 shows from thethat imaging the first, equation second, [and21]. third lens groups are fixed. Therefore, the collimatorBefore system the lens can group be conveniently moves in the used collimator. On the optical other system, hand, when the collimated the second beam lens is groupincident moves to the by testΔf, the lens, magnification and the magnification is calculated of in the the collimator same manner optical as that system in Equation becomes (1);infinite. thus, it If is the expressed magnification by Equation is calculated (3). using the Gaussian bracket at the refracting

power and interval of each lens group, the result is expressed1 u in3 Equation (1) [22,23]. In []−−−Δ−+Δ==zk,, z fk ,, z fk , (3) Equation (1), minf denotes01 the 1 magniﬁcation 2 2 when 3 themu collimated beam is incident to the test lens, and this value is inﬁnite. z and k are the same asCL the values0 shown in Figure2. Here, mCL may be expressed as angle u0 of an axial ray starting from the target and angle u3 of an axial ray incident to the test lens when the1 second lens group moves. In [−z0, k1, −z1, k2, −z2, k3] = = 0 (1) addition, the virtual target created by the collimator systemminf is imaged again by the test In addition, the refractive power KCL of the collimator system can be expressed by Equation (2), using the Gaussian bracket.

[k1, −z1, k2, −z2, k3] = KCL (2)

Figure2 shows that the first, second, and third lens groups are fixed. Therefore, the collimator system can be conveniently used. On the other hand, when the second lens group moves by ∆f, the magnification is calculated in the same manner as that in Equation (1); thus, it is expressed by Equation (3).

1 u3 [−z0, k1, −z1 − ∆ f , k2, −z2 + ∆ f , k3] = = (3) mCL u0

Here, mCL may be expressed as angle u0 of an axial ray starting from the target and angle u3 of an axial ray incident to the test lens when the second lens group moves. In addition, the virtual target created by the collimator system is imaged again by the test lens Appl. Sci. 2021, 11, 3350 5 of 11

Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 12 into an image sensor, and the magniﬁcation of the test lens is calculated from the imaging equation. The result is calculated using Equation (4).

lens into an image sensor, and the magnificationu of∆ thel test lens is calculated from the m = 3 = (4) imaging equation. The result is calculated usingu4 Equationf (4). Δ u3 l Finally, the magnification of them == optical system, including the collimator optical(4) system uf and the entire test lens, is denoted as m4T, which is calculated using Equation (5). Finally, the magnification of the optical system, including the collimator optical system and the entire test lens, is denotedu0 as muT0, whichu3 is calculatedf using Equation (5). mT = = = mCLm = (5) uuuu4 u3 u4 fCL ==003 = =f mmmTCL (5) Here, as shown in Figureuuu2,434 because u0 and u4 do fCL not significantly change according to the movementHere, as shown of the in second Figure lens 2, because group, um0T andalso u4 hardly do not changes. significantly Therefore, change Equationaccording (6) can beto obtainedthe movement by substituting of the second Equations lens group, (4) m andT also (5) hardly into Equation changes. (3) Therefore, and rearranging Equation them. (6) can be obtained by substituting Equations (4) and (5) into Equation (3) and rearranging 1 u m ∆l f ∆l them. [− − − − + ] = = 3 = = = CL z0, k1, z1 ∆ f , k2, z2 ∆ f , k3 2 (6) mCL u0 ΔmT mTΔf f 1 ufl3 mlCL []−−−Δ−+Δ=====zk,, z fk ,, z fk , (6) Therefore,01 the 1 refracting 2 2 power of 3 each lens group can be obtained2 by solving Equa- mummffCL0 T T tions (1), (2), and (6) in series. Here, we have explained that f is the focal length of the test Therefore, the refracting power of each lens group can be obtained by solving Equa- lens, which is determined as the reciprocal of k in Figure2. ∆l is also a determined value. tions (1), (2), and (6) in series. Here, we have explained that f is the focal length of the test Inlens, addition, which is if determined the motion as distance the reciprocal∆f of the of k second in Figure lens 2. groupΔl is also and a thedetermined value between value. each lensIn addition, group are if the properly motion determined,distance Δf of onlythe second unknown lens group variables and thek1, kvalue2, and betweenk3 remain. each lens Distancegroup arez properly0 from the determ targetined, to group only unknown 1 is 5 mm. variables Distance k1, zk12, fromand k the3 remain. first to the second lens groupDistance is 40z0 from mm. the Distance target toz2 groupfrom the1 is second5 mm. Distance to the third z1 from lens the group first to is the 30 second mm. Motion distancelens group∆f isof 40 the mm. second Distance lens z2 group from the is20 second mm. to Focal the third length lens fof group thetest is 30 lens mm. is Motion 4 mm. If the totaldistance focal Δ lengthf of the secondfCL of the lens collimator group is 20 optical mm. Focal system length isset f of to the 100 test mm, lens ∆isl 4becomes mm. If the 0.1 mm. Wetotal substitute focal length this fCL value of the intocollimator Equations optical (1), system (2), andis set (6). to 100 Thereafter, mm, Δl becomesf 1(=1/ k0.11), mm.f 2(=1/ k2), andWe fsubstitute3(=1/k3) this can value be obtained into Equati usingons the (1), MATLAB (2), and (6). programming Thereafter, f1(=1/ codes.k1), f2(=1/k2), and f3(=1/Ink3) Figure can be3 obtained, f 1, f 2, and usingf 3 therepresent MATLAB the programming focal lengths andcodes. the reciprocal of the refracting powerIn of Figure each 3, lens f1, f2 group., and f3 represent Figure3 shows the focal that lengths three and solutions the reciprocal can be obtained.of the refracting In the first solution,power of theeach focal lens lengthgroup. ofFigure the first3 shows lens that group three is solutions very short, can and be obtained. in the third In the solution, first the focalsolution, length the of focal the length second of lensthe first group lens is group very short.is very Thus, short, theand optically in the third meaningful solution, the solution becomesfocal length the of second the second solution. lens group The focalis very length short. ofThus, each the lens optically group meaningful can be determined solution from becomes the second solution. The focal length of each lens group can be determined from the obtained solution. Assuming that these lens groups have no aberration, the paraxial the obtained solution. Assuming that these lens groups have no aberration, the paraxial layout is shown in Figure4. layout is shown in Figure 4.

Figure 3. Result from solving Equations (1), (2), and (6) in a series of solutions using MATLAB codes. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 12

Appl. Sci. 2021, 11, 3350 6 of 11 Figure 3. Result from solving Equations (1), (2), and (6) in a series of solutions using MATLAB codes.

The 2nd lens group

The 3rd lens group The 1st lens group

Infinity

focusing

Near

FigureFigure 4. 4. ResultResult of of the the paraxial paraxial power power arrangement arrangement of of the the focus-adjustable focus-adjustable collimator collimator system. system.

3.3. Results Results and and Discussion Discussion WhenWhen the the refractive refractive power arrangement isis completed,completed, as as shown shown in in Figure Figure4, the4, the actual ac- tuallens lens must must be installed be installed to the to employed the employed optical optical system. system. In this In case, this the case, module the module lens method lens methodis utilized is utilized [24], which [24], iswhich the method is the method used to used determine to determine the curvature the curvature and thickness and thick- of a nesslens of that a lens minimizes that minimizes the third-order the third-orde aberrationr aberration of each group, of each using group, approximately using approxi- two to matelythree lensestwo to in three each lenses group. in Ineach this group. manner, In this foran manner, optical for system an optical where system the actual where lens the is actualapplied, lens an is optimizationapplied, an optimization process must process be performed must be usingperformed the optical using designthe optical software design to softwaremaximize to themaximize overall the resolution overall performance.resolution performance. In addition, In an addition, equivalent-lens an equivalent-lens conversion conversionmethod is usedmethod when is used a lens when is added a lens to is further added correctto further the correct aberrations the aberrations [25,26]. [25,26]. TheThe result result of of the the optical optical system system designed in our proposed methodmethodis is shown shown in inFigure Figure5 . 5.Figure Figure5a 5a shows shows the the optical optical layout layout of of our our designed designed collimatorcollimator systemsystem forfor testingtesting the resolutionresolution performance performance of of a atest test lens lens at at infi infinity.nity. Figure Figure 5b5b shows the optical layout for testingtesting the the test test lens lens at at an an object object distance distance of of approximately approximately 160 160 mm. mm. Figure Figure 55 showsshows that thethe first first lens lens group group consists consists of of three three lenses. lenses. In In addition, addition, two two lenses lenses are are used used in in the the second second andand third third lens lens group. group. In In the the optical optical system, system, shown shown in in Figure Figure 5,5, the the focal focal length length of of the the first first − lenslens group group is is approximately approximately −289.435289.435 mm, mm, that that of of the the second second lens lens group group is is approximately approximately − −61.93161.931 mm, mm, and and that that of ofthe the third third lens lens group group is is51.260 51.260 mm. mm. In addition, In addition, the theamount amount of of movement of the second lens group is approximately 17.617 mm. In Figure5b, H1G1 movement of the second lens group is approximately 17.617 mm. In Figure 5b, H1G1 rep- represents the distance from the first surface of the first lens group to the first principal resents the distance from the first surface of the first lens group to the first principal plane plane of the first lens group, and H2G1 represents the distance from the second principal of the first lens group, and H2G1 represents the distance from the second principal plane of plane of the first lens group to the last surface of the first lens group. Similarly, H1G2 the first lens group to the last surface of the first lens group. Similarly, H1G2 and H2G2 denote and H denote the distances from the first surface of the second lens group to the first the distances2G2 from the first surface of the second lens group to the first principal plane of principal plane of the first lens group and that from the second principal plane of the second the first lens group and that from the second principal plane of the second lens group to lens group to the last surface of the first lens group, respectively. Similarly, H1G3 and H2G3 denote the distances from the first surface of the third lens group to the first principal plane of the first lens group and that from the second principal plane of the second lens group to Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 12

Appl. Sci. 2021, 11, 3350 7 of 11 the last surface of the first lens group, respectively. Similarly, H1G3 and H2G3 denote the distances from the first surface of the third lens group to the first principal plane of the first lens group and that from the second principal plane of the second lens group to the thelast last surface surface of ofthe the first ﬁrst lens lens group, group, respectively. respectively. The The principal principal plane plane of of each each lens lens group group is isshown shown in in Figure Figure 5b.5b.

17 1 1 Perfect Lens 15 (f=4.0mm) The 1st lens group 6 6 H =7.572 1G2 HFOV=45° 7 The 2nd lens group 7 10 10 The 3rd lens group 11 11 14 14 H1 H2 H1 H2 H1 H2 H =43.472 17 1517 2G1 15

H2G3=9.001 H1G1=6.928 H =0.917 30.0 MM 1G3

H2G2=1.571 Focusing

(a) Position 1 (infinity) (b) Position 2 (Near)

Figure 5. Layout of the aberration-balanced collimator system at inﬁnity (Position 1) and the close Figure 5. Layout of the aberration-balanced collimator system at infinity (Position 1) and the close position (Position 2). position (Position 2). To commercialize our designed collimator system for mobile camera phone optics, we needTo tocommercialize obtain a relational our designed expression collimator for the displacementsystem for mobile of the camera second phone lens group optics, andwe need the defocus to obtain of a the relational test lens. expression When the for test the lensdisplacement is installed of behindthe second the designedlens group collimator,and the defocus the equation of the fortest the lens. height When of the axialtest lens ray is expressedinstalled behind in Gaussian the designed brackets, colli- as inmator, Equation the equation (7) [27,28 ].for the height of the axial ray is expressed in Gaussian brackets, as in Equation (7) [27,28]. [−−−Δ−+Δ−−−Δ=[−z0, k1, −z1 − ∆ f , k2, −z2 + ∆ f , k3, −z3, k, − f − ∆]l] = 0 (7) zk01,, z 1 fk ,, 2 z 2 fk ,, 3 zk 3 ,, f l 0 (7) InIn Equation Equation (7), (7),∆ Δf frepresents represents the the motion motion amount amount of of the the second second lens lens group, group, and and∆ lΔl denotesdenotes the the defocus defocus of of the the test test lens. lens. In In addition, addition, when when the the crane crane design design is is completed, completed, the the refractingrefracting power power and and spacing spacing of of each each lens lens group group are are determined. determined.z 3z3,, which which has has not not been been previouslypreviously mentioned, mentioned, denotes denotes the the distance distance between between the the last last group group of of the the collimator collimator and and thethe test test lens. lens. TheThe magniﬁcation, magnification,m TmT,, calculated calculated using using Equation Equation (5), (5), can can be expressed be expressed as Equation as Equation (8) using(8) using the Gaussianthe Gaussian bracket bracket [29]. [29]. 11 f f []−−−−==[− − − − ] = = CLCL zk01,,z0, k1, zk 12 ,,z1, k2, zk 23z ,,2, k3, zk 3z ,3, k (8)(8) mT f mfT

ExceptExcept forforz 3z3in in Equation Equation (8), (8), the the values values of of the the other other variables variables are are already already known. known. Therefore,Therefore,z 3z3can can be be obtained obtained from from Equation Equation (8). (8). Hence, Hence, the the unknown unknown variables variables in in Equa- Equa- tiontion (7) (7) are are∆ Δf fand and∆ Δl.l. In In fact, fact, in in the the optical optical system system designed designed and and shown shown in in Figure Figure5, 5,z3 zis3 is approximatelyapproximately 21.001 21.001 mm. mm. The The equations equations for for both both variables variables can can be be found found using using MATLAB MATLAB codes.codes. The The values values in in the the graph graph are are shown shown in in Figure Figure6. 6. In In addition, addition, the the values values of of the the locus locus ofof the the ﬁnal final design design using using Equation Equation (7) (7) are are listed listed in in Table Table1. 1. Figure Figure6a shows6a shows the the graph graph of of thethe ﬁrst first and and second second columns columns of of Table Table1. 1. Figure Figure6b 6b shows shows the the graph graph of theof the third third and and fourth fourth columnscolumns of of Table Table1. 1. Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 12 Appl. Sci. 2021, 11, 3350 8 of 11

(a) the case where the lens group (b) the case where the image moves linearly plane moves linearly

Figure 6. Locus of the second lens group of the focus-adjustable collimator. Figure 6. Locus of the second lens group of the focus-adjustable collimator. Table 1. Zoom locus of the focus-adjustable collimator. Table 1. Zoom locus of the focus-adjustable collimator. dl Df df dl dl 0.0Df 0.000000 df 0.00 0.000000 dl 0.0 2.00.000000 0.009163 0.00 0.01 2.177721 0.000000 2.0 4.00.009163 0.018806 0.01 0.02 4.240659 2.177721 6.0 0.028958 0.03 6.199488 4.0 0.018806 0.02 4.240659 8.0 0.039654 0.04 8.063044 6.0 10.00.028958 0.050930 0.03 0.05 9.839028 6.199488 8.0 12.00.039654 0.062829 0.04 0.06 11.534208 8.063044 10.0 14.00.050930 0.075398 0.05 0.07 13.154574 9.839028 16.0 0.088687 0.08 14.705465 12.0 18.00.062829 0.102754 0.06 0.09 16.191659 11.534208 14.0 20.00.075398 0.117663 0.07 0.10 17.617460 13.154574 16.0 0.088687 0.08 14.705465 18.0Figure 7 shows0.102754 the MTF curve of the collimator 0.09 designed and shown in Figure 16.1916595. The MTF20.0 graph must0.117663 be utilized to verify the optical 0.10 system performance because 17.617460 the MTF represents the optical aberrations and resolutions of the optical systems [30–33]. Figure7 showsFigure that 7 theshowsx-axis the of eachMTF graph curve represents of the collimator the movement designed of the image and sensorshown assembled in Figure 5. The MTFon graph the test must lens, andbe utilized this represents to verify the defocus.the optical The ysystem-axis represents performance the MTF because values of the MTF representsthe designed the optical optical system.aberrations MTF is and defined resoluti as theons value of obtainedthe optical by dividing systems the [30–33]. visibility Figure 7 showsof the that image the planex-axis by of that each of graph the object represents plane [34 ,the35]. movement If this value isof high, the image the resolution sensor assem- bledis on sufficiently the test lens, high toand commercialize this represents the opticalthe defocus. system The product y-axis in therepresents factory [36 the,37 MTF]. In values addition, the solid line represents the MTF in the tangential direction, and the dotted of the designed optical system. MTF is defined as the value obtained by dividing the vis- line represents that in the sagittal direction. If the image sensor was properly assembled, ibilitythe of image the image quality plane would by appear that very of the clear object for commercialization. plane [34,35]. If However,this value at is an high, object the reso- lutiondistance is sufficiently of 16 cm, the high resolution to commercialize at the periphery the is expectedoptical tosystem be low. product In fact, when in the the factory [36,37].object In distance addition, from the the solid mobile line phone represents camera is 16the cm, MTF the resolutionin the tangential is supposed direction, to be low and the dottedbecause line ofrepresents the out-of-focus that conditionin the sagittal at the periphery.direction. Thus, If the a levelimage that sensor can be sufficientlywas properly assembled,used in the a general image mobile quality phone would camera appear should very be found.clear for commercialization. However, at Figure8 shows the lateral color of the optical system as shown in Figure5. As shown an object distance of 16 cm, the resolution at the periphery is expected to be low. In fact, in Figure8, the maximum value of the lateral color is from −1.0 to −0.63 µm. These values whencorrespond the object to thedistance size of onefrom pixel the of mobile the image ph sensor.one camera Therefore, is chromatic16 cm, the aberration resolution due is sup- posedto theto be movement low because of the focusingof the out-of-focus lens is not a problem.condition For at this the reason, periphery. we can Thus, conclude a level that can thatbe sufficiently it has a stable used resolution in a general performance, mobile as shownphone in camera the MTF should curve in be Figure found.7. Figure 8 shows the lateral color of the optical system as shown in Figure 5. As shown in Figure 8, the maximum value of the lateral color is from −1.0 to −0.63 µm. These values correspond to the size of one pixel of the image sensor. Therefore, chromatic aberration due to the movement of the focusing lens is not a problem. For this reason, we can conclude that it has a stable resolution performance, as shown in the MTF curve in Figure 7. Appl. Sci. 2021, 11, 3350 9 of 11 Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 12

Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 12

Diffraction MTF FCL inf 1

0.9 Diffraction MTF FCL inf 0.8 1 F1: Diff. Limit 0.7 0.9 F1: (RIH) 0.000 mm F2: T (RIH) 1.400 mm 0.6 0.8 F2: R (RIH) 1.400 mm F1:F3: Diff. T Limit (RIH) 2.400 mm 0.5 F1: (RIH) 0.000 mm 0.7 F3: R (RIH) 2.400 mm F2: T (RIH) 1.400 mm F4: T (RIH) 3.200 mm 0.4 0.6 F2: R (RIH) 1.400 mm F4: R (RIH) 3.200 mm F3: T (RIH) 2.400 mm 0.3 0.5 F3:F5: R (RIH) T (RIH) 2.400 4.000 mm mm F5: R (RIH) 4.000 mm 0.2 0.4 F4: T (RIH) 3.200 mm F4:F6: R (RIH) T (RIH) 3.200 4.100 mm mm 0.1 0.3 F5:F6: T (RIH) R (RIH) 4.000 4.100 mm mm F5: R (RIH) 4.000 mm 0 0.2 F6: T (RIH) 4.100 mm -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01F6: R (RIH) 4.100 mm 0.1 Defocusing Position (mm) 0 -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01 (a) ThroughDefocusing focus Position MTF (mm) plots at infinity (a) Through focus MTF plots at infinity Modulation Modulation

(b) Through focus MTF plots at 60cm (b) Through focus MTF plots at 60cm Modulation Modulation

(c)(c) Through Through focusfocus MTF plots atat 16cm16cm

FigureFigure 7. 7.ModulationModulation Modulation transfer transfer function function (MTF) (MTF) grap graphh of of thethe aberration-balancedaberration-balanced collimator. collimator.

C-line C-line C-line C-line C-line C-line FCL inf d-line FCL 60cm d-line FCL 20cm d-line FCL inf d-line FCL 60cm d-line FCL 20cm d-line e-line e-line e-line e-line e-line e-line F-line F-line F-line F-lineg-line g-lineF-line g-lineF-line g-line g-line g-line

1.0 1.0 1.0 1.0 1.0 1.0

0.9 0.9 0.9 0.9 0.9 0.9

0.8 0.8 0.8 0.8 −1.00μm at 0.85-field 0.8 −0.80μm at 0.85-field 0.8 0.7 −1.00μm at 0.85-field 0.7 −0.80μm at 0.85-field 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 −0.63μm at 0.65-field 0.5 0.5 0.5 −0.63μm at 0.65-field 0.5 0.5 0.5

Field Ratio Field 0.4 Ratio Field 0.4 Ratio Field 0.4

Field Ratio Field 0.4 Ratio Field 0.4 Ratio Field 0.4 0.3 0.3 0.3

0.3 0.3 0.3 0.2 0.2 0.2

0.2 0.2 0.2 0.1 (a) infinity 0.1 (b) 60cm 0.1 (c) 20 cm 0.1 0.1 0.1 0.0 0.0 0.0 (a)-1.2 infinity -1.0 -0.7 -0.5 -0.2 0.0 0.2 0.5 0.7 1.0 1.2 -1.2 (b) -1.0 60cm -0.7 -0.5 -0.2 0.0 0.2 0.5 0.7 1.0 1.2 -1.2(c) -1.0 -0.720 -0.5 cm -0.2 0.0 0.2 0.5 0.7 1.0 1.2 Lateral Color(um), LSF Lateral Color(um), LSF Lateral Color(um), LSF 0.0 0.0 0.0 -1.2 -1.0 -0.7 -0.5 -0.2 0.0 0.2 0.5 0.7 1.0 1.2 -1.2 -1.0 -0.7 -0.5 -0.2 0.0 0.2 0.5 0.7 1.0 1.2 -1.2 -1.0 -0.7 -0.5 -0.2 0.0 0.2 0.5 0.7 1.0 1.2 Lateral Color(um), LSF Lateral Color(um), LSF Lateral Color(um), LSF FigureFigure 8. 8.Lateral Lateral colorcolor of our designed collimator. collimator. Figure 8. Lateral color of our designed collimator. Appl. Sci. 2021, 11, 3350 10 of 11

Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 12

The fabrication result of the designed optical system is shown in Figure9. Figure5 showsThe that fabrication the resolution result chartof the anddesigned optical optica systeml system are locatedis shown at in a shortFigure distance. 9. Figure Therefore,5 whenshows an that optical the resolution system is chart assembled, and optical it must system be assembledare located usingat a short a resolution distance. chart.There- Because fore, when an optical system is assembled, it must be assembled using a resolution chart. the position of the virtual image must be changed, the motor is also assembled. Because the position of the virtual image must be changed, the motor is also assembled.

FigureFigure 9. ManufacturingManufacturing result result of the of theoptical optical system system shown shown in Figure in Figure 5. 5.

4.4. ConclusionsConclusions Measuring the the resolution resolution of an of anoptical optical system system is commonly is commonly performed performed using an using ISO an ISO 12,23312,233 resolution chart. chart. This This method method requires requires the resolution the resolution chart to chart be fixed to be at fixed a specific at a specific position.position. This This measurement measurement method method requires requires a lot of aspace. lot of For space. this reason, For this a small reason, reso- a small resolutionlution chartchart is conventionally is conventionally placed at placed a short atdistance, a short and distance, a collimator and system a collimator is placed system is placedbetween between the resolution the resolution chart and charttest lens. and When testlens. this is When performed, this is a performed,virtual image a with virtual a image resolution chart enlarged by a collimator is created. This method can save a lot of meas- with a resolution chart enlarged by a collimator is created. This method can save a lot urement space compared with the method that directly uses a resolution chart. However, of measurement space compared with the method that directly uses a resolution chart. because the resolution chart is also fixed in this method, measuring the change in the res- However,olution according because to the the resolution change in the chart object is also distance fixed is in not this possible. method, Therefore, measuring in our the change instudy, the resolutiona focus-adjustable according collimator to the system change was in designed. the object By distance using this is developed not possible. optical Therefore, insystem, our study, a method a focus-adjustable of measuring the collimator resolution system according was to designed.the change By in usingthe object this dis- developed opticaltance while system, reducing a method the measurement of measuring space the resolutionwas proposed. according In addition, to the we change confirmed in the object distancethat the MTF while performance reducing theof the measurement collimator itself space could was beproposed. sufficientlyIn ensured addition, even we if the confirmed thatlens theinside MTF the performancecollimator moved. of the collimator itself could be sufficiently ensured even if the lens inside the collimator moved. Author Contributions: Conceptualization, H.C., S.-w.C. and J.R.; methodology, H.C. and J.R.; formal analysis, J.R.; writing—original draft preparation, H.C., S.-w.C. and J.R. All authors have read Author Contributions: Conceptualization, H.C., S.-w.C. and J.R.; methodology, H.C. and J.R.; formal and agreed to the published version of the manuscript. analysis, J.R.; writing—original draft preparation, H.C., S.-w.C. and J.R. All authors have read and agreedFunding: to theThis published work was versionsupported of by the the manuscript. National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C4001606). This research was also sup- Funding:ported by ThisBasic workScience was Research supported Program by thethrough National the National Research Research Foundation Foundation of Korea of Korea (NRF) grant funded(NRF) funded by the by Korea the Ministry government of Education (MSIT) (No.(NRF2020R1I1A3052712). 2020R1A2C4001606). This This research research was was supported also supported byby Basicthe National Science Research Research Foundation Program of Korea through (NRF) the grant National funded Researchby the Korea Foundation government of(MSIT) Korea (NRF) funded(NRF-2019R1F1A1062397). by the Ministry of Education (NRF2020R1I1A3052712). This research was supported by theInstitutional National Review Research Board Foundation Statement: of Not Korea applicable. (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1062397). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are included within the article. Appl. Sci. 2021, 11, 3350 11 of 11

Conﬂicts of Interest: The authors declare no conﬂict of interest.

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