applied sciences

Article 3D Point Spread Function Evaluation Method on a Confirmed Object Plane Perpendicular to the Optical Axis

Shuai Mao *, Zhenzhou Wang and Jinfeng Pan School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo 255000, China; [email protected] (Z.W.); [email protected] (J.P.) * Correspondence: [email protected]

 Received: 22 January 2020; Accepted: 1 April 2020; Published: 2 April 2020 

Featured Application: A point spread function (PSF) evaluation method of a microscope on the object plane that is perpendicular to the optical axis is proposed. The proposed method can realize the 3D PSF on the perpendicular object plane, magnification, and paraxial region evaluation, and it can confirm any microscopic system.

Abstract: A point spread function evaluation method for a microscope on the object plane that is perpendicular to the optical axis is proposed. The measurement of the incident beam direction from the dual position-sensitive-detector (PSD)-based units, the determination of the object plane perpendicularity and the paraxial region, and evaluation methods for the point spread function (PSF) are presented and integrated into the proposed method. The experimental verification demonstrates that the proposed method can achieve a 3D PSF on the perpendicular object plane, as well as magnification, paraxial region evaluation, and confirmation for any microscopic system.

Keywords: point spread function; microscope; optical axis; object plane

1. Introduction In microscopic imaging, the imaging of a point-like light source does not appear as a point due to a diffraction pattern termed the point spread function (PSF). The PSF describes the analytical ability of the microscopic imaging of a point source and is a significant parameter for evaluating the resolution and quality of microscopic imaging systems. Therefore, an accurate measurement of the PSF is critical for the evaluation of microscopic systems [1,2]. The accurate measurement of the PSF is especially important when using super-resolution microscopic technologies [3,4]. For example, stochastic optical reconstruction and photo-activated localization microscopy both need accurate PSFs to fit each fluorescent point so that the center of each point can be exactly located [5–7]. Furthermore, structured illumination microscopy needs accurate PSFs to build the correct optical transfer function for image restoration and to obtain super-resolution images [8–11]. There are usually two ways to obtain a PSF: numerical calculations and experimental measurements. Numerical calculations require the PSF mode of theory because the difficult estimation of the pupil function often results in the theoretical model not reflecting the real imaging system [12,13]. Therefore, experimental measurements are the preferred method to obtain PSFs for microscopic systems. The most common methods for experimentally obtaining PSFs are the star point detection, knife-edge, and fluorescent microsphere detection methods [13,14]. The PSF for microscopic imaging is commonly obtained with the fluorescent microsphere detection method. The PSF should be measured on a pre-determined object plane that is perpendicular to the optical axis of the microscopic system (Figure1). The PSF has a correspondence relationship with an object

Appl. Sci. 2020, 10, 2430; doi:10.3390/app10072430 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 2430 2 of 13

Appl. Sci. 2020, 1, x FOR PEER REVIEW 2 of 13 plane that is perpendicular to the optical axis, and a 3D PSF measurement is used to obtain each PSFeach on PSF its oncorresponding its corresponding perpendicular perpendicular object plane object at plane the focus at the depth. focus Realizing depth. R thisealizing goal this becomes goal morebecomes diffi morecult with difficult super-resolution with super-resolution microscopic microscopic measurements. measurements. A measured A measured object plane object that isplane not perpendicularthat is not perpendicular to the optical to axis the resultsoptical in axis an obtainedresults in PSF an thatobtained is not PSF homogeneous that is not duehomogeneous to a defocusing due etoff ecta defocusing along with effectthe optical along axis with direction. the optical Moreover, axis direction. the accuracy Moreover, of the PSF the is accuracy significantly of the reduced PSF is whensignificantly the PSF reduced does not when have the a correspondingPSF does not have object a corresponding plane that is perpendicularobject plane that to is the perpendicular optical axis. Thisto the is opt especiallyical axis. important This is especially when the important PSF is employed when the in PSF 3D super-resolutionis employed in 3D microscopic super-resolution image restoration.microscopic Inimage this case,restoration. all the correspondingIn this case, all objectthe corresponding planes within object focal depthplanes mustwithin be focal confirmed. depth Twomust other be confirmed. microscopic T elementswo other are microscopic essential when elements obtaining are a PSF:essential the magnification when obtaining and the a PSF: paraxial the region.magnification These elementsand the paraxial may change region. with These object elements plane variation may change along with the object optical plane axis. variation In summary, along if thethe measuredoptical axis. object In summary, plane that if is the perpendicular measured object to the plane optical that axis is perpendicular is not determined, to the the optical meanings axis ofis thenot PSF, determined, magnification, the meanings and paraxial of regionthe PSF, will magnification, be lost for super-resolution and paraxial microscopic region will measurements. be lost for Thesuper common-resolution methods microscopic for obtaining measurements. PSFs stated The above common are not methods able to determinefor obtaining the PSFs object stated planes above that are perpendicular not able to determine to the optical the axis object or the planes longitudinal that are height perpendicular positions of to each the object optical plane axis along or the opticallongitudinal axis, asheight measuring positions these of factorseach object requires plane more along than the 3D optical PSF measurements. axis, as measuring these factors requires more than 3D PSF measurements.

Image plane Ideal imaging region corresponding to the paraxial region of one perpendicularity object plane

Optical system

Microscope

The measured Object plane Optical Axis Figure 1. The measured object plane is perpendicular to the optical axis of the microscopic system.system.

InIn this this paper, paper, a a microscopemicroscope PSF PSF evaluation evaluation method method forfor calibratingcalibrating the object object plane thatthat is is perpendicularperpendicular to the the optical optical axis axis is is proposed. proposed. The paraxialparaxial region,region, objectobject plane plane perpendicularity perpendicularity determination,determination, andand 3D 3D PSF PSF evaluation evaluation methods methods are are presented. presented. The The presented presented methods, methods, together together with thewith integration the integration of dual-spot of dual position-spot detectionposition detection technology technology to realize a to 3D realize PSF on a the 3D perpendicular PSF on the objectperpendicular plane, facilitate object plane, the magnification facilitate the and magnification paraxial region and evaluation,paraxial region as well evaluation as the confirmation, as well as the of anyconfirmation microscopic of system.any microscopic The availability system. of the The proposed availabil methodity of is thethen proposed experimentally method verified. is then experimentally verified. 2. Layout of PSF Measurement 2. LayoutThe proposed of PSF Measurement PSF measurement configuration consisted of a movable assembly (PSF calibration module)The and proposed a static assembly PSF measurement (light source configuration and Z-axis position consist measuremented of a movable charge-coupled assembly devices (PSF (CCDs);calibration Figure module)2). The andcore component a static assembly was a dual (light position-sensitive-detector source and Z-axis position (PSD)-based measurement unit [ 15] thatcharge was-coupled within the devices PSF calibration (CCD module.s); Figure The dual2). PSD-based The core unit was component used to obtainwas the a incident dual beamposition direction-sensitive vector.-detector Here, (PSD) two PSDs-based were unit replaced [15] that by wa twos within CCDs th thate PSF could calibration obtain two module. light spots The fromdual twoPSD- dibasedfferent unit light wa routes.s used Into thisobtain way, the the incident dual PSD-based beam direction unit was vector. able toHere, obtain two two PSDs incident were beamreplaced light by direction two CCDs vectors that could at the obtain same two time. light When spots a light from beam two different reached onelight receiving routes. In plane, this way the, intensitythe dual PSD distribution-based unit of the wa lights able spot to obtain acted astwo a form incident of a Gaussianbeam light surface, direction and vectors the plane at the position same oftime. the When light spota light was beam directly reache obtainedd one byreceiving the PSD. plane The, intensitythe intensity distribution distribution of the of light the light spot spot was imagedacted as bya form the CCD, of a Gaussian and a Gaussian surface function, and the wasplane applied position to of fit the light intensity spot distribution;was directly then,obtained the planeby the position PSD. The was intensity obtained distribution by fitting. of the light spot was imaged by the CCD, and a Gaussian function was applied to fit the intensity distribution; then, the plane position was obtained by fitting. Appl. Sci. 2020, 10, 2430 3 of 13 Appl. Sci. 2020, 1, x FOR PEER REVIEW 3 of 13

Movable assembly Microscope Quarter wave plate Fluorescent particle

Beam shifting CC

Polarization Splitter plane Polarization refractor plane Static assembly Rectangular prism Dual PSD-based unit

CCD

PSF calibration module

Outgoing Incident six-degree-freedom Z beam light beam light displacement X Rolling Pitching Y FigureFigure 2. 2.Proposed Proposed point point spread spread function function (PSF) (PSF measurement) measurement setup setup and light and routes light routes (red/purple (red/purple arrows). arrows). The PSF calibration module was integrally fixed on a six-degree-freedom macro/micro displacement platform.The PSF The slopecalibration face of module the rectangular was integrally prism containing fixed on the a six polarization-degree-freedom refractor macro/micro plane and polarizationdisplacement splitter platform. plane The was slope parallel face toof the Y–Zrectangular plane of prism the coordinates, containing asthe determined polarization by refractor the dual PSD-basedplane and polarization unit. The essential splitter characteristicsplane was parallel of the to proposedthe Y–Z plane PSF measurementof the coordinates setup, as included determined the followingby the dual (Figure PSD-2based): unit. The essential characteristics of the proposed PSF measurement setup 1.includeA linearlyd the following polarized (Fig beamure of2): light originated from a fiber-coupled laser (mounted on an adjustor).

1.TheA linearly linearly polarized polarized beam beam light of light could originate go throughd from the a polarization fiber-coupled refractor laser (mounted plane and on was an adjustor). The linearly polarized beam light could go through the polarization refractor divided in half by the polarization splitter plane. plane and was divided in half by the polarization splitter plane. 2. The light beam from the laser was the excitation light of a fluorescent particle. The cover slip 2. The light beam from the laser was the excitation light of a fluorescent particle. The cover slip acted as an emission filter [16] that only allowed the fluorescent particle’s emission light to travel acted as an emission filter [16] that only allowed the fluorescent particle’s emission light to into the microscope. travel into the microscope. 3. 3. WhenWhen the the linearly linearly polarized polarized light light beam beam reached reache thed the Z-position Z-position measurement measurement CCD CCD (mounted (mounted on anon adjustor) an adjustor) from the from beam the shifting beam shifting corner cube corner (CC), cube after (CC), going after through going the through quarter the wave quarter plate, it changedwave plate, into it circularly changed polarizedinto circularly light. polarized A part of thislight. light A part was of reflected this light by thewas semi-transparent reflected by the mirrorsemi- paralleltransparent to the mirror Z-position para measurementllel to the Z CCD’s-position light-sensitive measurement surface. CCD’s Another light- partsensitive was directedsurface. into Another the Z-position part wa measurements directed into CCD. the When Z-position the reflected measurement part again CCD.went through When the quarterreflect waveed part plate, again its linearly went through polarized the direction quarter was wave perpendicular plate, its linearly to the original. polarized This direction portion waswas reflected perpendicular by the polarization to the original. refractor This plane portion and wa directeds reflect intoed theby the dual polarization PSD-based unit.refractor plane and directed into the dual PSD-based unit. 3. Core Technologies and Algorithms 3. Core Technologies and Algorithms 3.1. Measurement of the Direction of the Dual PSD-Based Unit’s Incident Beam 3.1. MeasurementDual PSD-based of the unitsDirection were of previously the Dual PSD proposed-Based Unit’s in [15 Incident]. In the Beam dual PSD-based unit proposed here,Dual the custom PSD-based beam units splitter were (BS) previously consisted proposed of two BSs, in both[15]. withIn the coplanar dual PSD reflecting-based unit surfaces. proposed The incidenthere, the lightcustom reflected beam bysplitter the custom (BS) consist BS traverseded of two a refractionBSs, both with offset coplanar piece as thickreflecting as the surfaces. first BS andThe thenincident entered light PSD reflected I. The by portion the custom of the BS incident traverse lightd a thatrefraction was transmitted offset piece through as thick the as the custom first BSBS wasand partlythen enter reflecteded PSD by I. a The CC portion and received of the byincident PSD II. light that was transmitted through the custom BS was partlyIn reflected Figure3, by it isa CC seen and that received the CC by joint PSD was II. located at the origin of the rectangular coordinate system.In Figure The cube’s 3, it centeris seen line that (i.e., the theCC Z-axis) joint wa ands located the custom at the BS’s origin reflecting of the surfaces rectangular were coordinate fixed at an anglesystem. of The 45◦. cube’sL1 is the center distance line (i.e. between, the Z the-axis) CC and joint the and custom the center BS’s reflecting point of the surface customs we BSre (i.e.,fixed the at point of intersection between the Z-axis and the custom BS’s reflecting surfaces). L and L are the an angle of 45°. L1 is the distance between the CC joint and the center point of the custom2 BS3 (i.e., the distances between the Z-axis and PSD I and PSD II, respectively. L and L represent the distances point of intersection between the Z-axis and the custom BS’s reflecting4 surface5 s). L2 and L3 are the between the CC joint and the center points of PSD I and II, respectively. The custom BS’s reflecting distances between the Z-axis and PSD I and PSD II, respectively. L4 and L5 represent the distances surfacesbetween andthe PSDsCC joint were and all the perpendicular center point tos theof PSD X–Z I plane, and II, assuming respectively. that (They1, z 1custom) and ( yBS’s2, z2 )reflecting were the coordinate values of the beam spots on PSD I and II, respectively. It is important to note that (y , z ) surfaces and PSDs were all perpendicular to the X–Z plane, assuming that (y1, z1) and (y2, z2) were1 the1 and (y , z ) are absolute coordinate values, whereas a PSD only measures its own values along its own coordinate2 2 values of the beam spots on PSD I and II, respectively. It is important to note that (y1, z1) Y- and Z-axes. Thus, z and z should actually be z + L and z + L in the rectangular coordinate and (y2, z2) are absolute1 coordinate2 values, whereas1P a PSD4 only2P measures5 its own values along its own Y- and Z-axes. Thus, z1 and z2 should actually be z1P + L4 and z2P + L5 in the rectangular coordinate system of Figure 3, respectively, where z1P and z2P are, respectively, the values measured by PSD I Appl. Sci. 2020, 1, x FOR PEER REVIEW 4 of 13

Appl.and Sci. II. 2020In addition,, 10, 2430 the centers of PSD I and II can deviate from the X–Z plane; here, they4 of are 13 expressed as L6 and L7, respectively. Thus, the values of y1 and y2 should be y1P + L6 and y2P + L7 in the Appl. Sci. 2020, 1, x FOR PEER REVIEW 4 of 13 rectangular coordinate system of Figure 3, respectively, where y1P and y2P are the values measured by systemPSD I and of Figure II. 3, respectively, where z1P and z2P are, respectively, the values measured by PSD I and and II. In addition, the centers of PSD I and II can deviate from the X–Z plane; here, they are II. In addition, the centers of PSD I and II can deviatePS fromD Ⅱ the X–Z plane; here, they are expressed as expressed as L6 and L7, respectively. Thus,L 5the values of y1 and y2 should be y1P + L6 and y2P + L7 in the L6 and L7, respectively. Thus, the values of y1 and y2 should be y1P + L6 and y2P + L7 in the rectangular rectangular coordinate system of Figure 3, respectively,R ewhereflected y1P and y2P are the values measured by coordinate system of Figure3, respectively, where y1P andsurfayce2 ⅡP are the values measured by PSD I and II. PSD I and II. L3 X CC PSD Ⅱ L5 Emergent light Custom BS 45° Reflected Y Z Reflescutrefdac e Ⅱ surface Ⅰ L3 CC X Incident ligEhmt ergent light L2 Custom BS 45° L1 Refraction Y Z ofRf-esfelte cptiedc e L4 surface Ⅰ PSD Ⅰ Incident light L2

Figure 3. Schematic of the designedL1 dual positionRefractio-sensitiven -detector (PSD)-based unit. off-set piece L4 In the rectangular coordinate system of FigurePSD Ⅰ 3, the components of the incident beam unit vector that arrive at the dual PSD-based unit are: Figure 3. Schematic of the designed dual position position-sensitive-detector-sensitive-detector (PSD)-based(PSD)-based unit. MF In the rectangular coordinate system of Figurem  32, the components of the incident beam unit vector In the rectangular coordinate system of Figure 3, the components of the incident beam unit that arrive at the dual PSD-based unit are: vector that arrive at the dual PSD-based MQunit are: n  1 (1) = M2 F m MF2∆− m  M1 Q n = E∆− (1) q −E , q = MQ∆ ,  n − 1 (1)  wherewhere 2L1 2+L1L+3L3−LL2=2E=, E, LL66++L7L=Q7 ,= Q,2 2LL11−L4−L45=F, L5 =y1FP,+y21PP=M+ 1y, 2P =z1MP+z12,P=z1MP 2+, z2Pand= M 2, and△= 2 q 2 − 2 − − √퐸 +(푀 − 푄) + (푀 − 퐹) . E, Q, and F are parametersE of the dual PSD-based unit, and M1 and M2 = E21 + (M Q)2 + (M F)2. E, Q, andq F are, parameters of the dual PSD-based unit, and M 4are the values obtained1 − by PSDs2 − I and II, respectively. The parameters E, Q, and F can be calibrated1 and M are the values obtained by PSDs I and II, respectively. The parameters E, Q, and F can be by the 2auto-reflection alignment method [17]. In addition, it is clear that the translation motion of calibratedwhere by2L1 the+L3− auto-reflectionL2=E, L6+L7= alignmentQ, 2L1− methodL4−L5=F, [ 17].y In1P+ addition,y2P=M1, it isz1P clear+z2P=M that2, theand translation △= the 2incident beam2 can be measured2 by PSD I and II in the dual PSD-based unit. motion√퐸 +(푀 of1 − the푄 incident) + (푀2 beam− 퐹) . can E, Q be, and measured F are parameters by PSD I and of the II in dual the dualPSD- PSD-basedbased unit, unit.and M1 and M2 3.2.are theParaxial values Region obtained and Object by PSDs Plane I and Perpendicularity II, respectively. Determination The parameters E, Q, and F can be calibrated 3.2.by the Paraxial auto- Regionreflection and alignment Object Plane method Perpendicularity [17]. In addition, Determination it is clear that the translation motion of the incidentAccording beam to thecan theorybe measured of diffraction, by PSD there I and exists II in the a paraxial dual PSD region-based of uniteach. perpendicular object According to the theory of diffraction, there exists a paraxial region of each perpendicular object plane that allows the nanoparticle’s imaging to satisfy an isoplanatic condition (Figure 4). In this plane that allows the nanoparticle’s imaging to satisfy an isoplanatic condition (Figure4). In this paraxial3.2. Paraxial region Region, the and aberrationsObject Plane Perpendicularity of the nanoparticle Determination’s imaging can be minimized, and the paraxial region, the aberrations of the nanoparticle’s imaging can be minimized, and the nanoparticle nanoparticle can be imaged ideally. Moreover, the images of the nanoparticles located in different can beAccording imaged ideally. to the theory Moreover, of diffraction, the images there of the exists nanoparticles a paraxial locatedregion of in each different perpendicular positions onobject the positions on the paraxial region of some perpendicular object planes are considered to be identical. It paraxialplane that region allows of somethe nanoparticle perpendicular’s imaging object planes to satisfy are considered an isoplanatic to be identical.condition It(Figure is clear 4) that. In there this is clear that there is one ideal imaging region corresponding to each of the paraxial regions on the isparaxial one ideal region imaging, the region aberrations corresponding of the to nanoparticle each of the paraxial’s imaging regions can on be the minimized imaging plane., and the imaging plane. nanoparticle can be imaged ideally. Moreover, the images of the nanoparticles located in different Optical Axis Ideal imaging region positions on the paraxial region of some perpendicular objectcor rplaneespondinsg areto pa consideredraxial region to be identical. It is clear that there is one Iidealmage imagingplane region corresponding to each of the paraxial regions on the imaging plane. Z Optical Axis Ideal imaging region corresponXdingR otoll paraxial region Image plane Pitch Y Perpendicular Z object plane PXaraxRioalll region Pitch Y Perpendicular Figure 4. Imaging of a nanoparticle on different object planes that are perpendicular to the optical axis. object plane Paraxial region

Appl. Sci. 2020, 1, x FOR PEER REVIEW 5 of 13

Appl. Sci.Figure2020 4., 10 Imaging, 2430 of a nanoparticle on different object planes that are perpendicular to the optical 5 of 13 axis.

Based on on the the di ff diffractionraction imaging imaging analysis, analysis, the spot the intensity spot intensity distribution distribution and size of and the nanoparticle size of the imagingnanoparticle change imaging with change the object with plane’s the object variation. plane’s However, variation. the However, intensity the distribution intensity distribution of these spot of imagesthese spot is very images symmetrical is very symmetrical when these images when are these within images the paraxialare within region the of paraxial each object region plane, of each and thoseobject thatplane are, and out t ofhose the that paraxial are out region of the are paraxial asymmetric region due are toasymmet an aberrationric due into thean aberration optical imaging in the system.optical imaging In Figure system.5, the curveIn Figure surface 5, wasthe curve fit with surface the nanoparticle’s was fit with intensitythe nanoparticle distribution’s intensity image withdistribut a Gaussianion image function with a to Gaussian obtain its function central position. to obtain Then, its central we divided position. the nanoparticle’sThen, we divide intensityd the distributionnanoparticle longitudinally’s intensity distributio into two partsn longitudinally by passing through into two the parts central by position passing (as through shown thein Figure central5, whereposition the (as splitting shown planein Figure is parallel 5, where to the splitting X–Z or Y–Z plane planes), is parallel where to thethe curveX–Z or surface Y–Z plane fit eachs), where of the twothe curve parts withsurface a Gaussian fit each of function. the two Next, parts we with normalized a Gaussian the function two fitting. Next, Gaussian we norm surfacesalize ofd the the two partsfitting of Gaussian the nanoparticle’s surfaces of intensity the two distribution. parts of the nanoparticle The full widths’s intensity at half maximum distribution (FWHM). The full of thewidths two normalizedat half maximum Gaussian (FWHM) surfaces of werethe two obtained, normalized and then Gaussian the di ffsurfaceerences between were obtained the two, FWHMsand then wasthe alsodifference obtained. between the two FWHMs was also obtained. Two divisions of intensity distribution of nano-particle

Gaussian Gaussian fitting fitting FWHM FWHM

Intensity distribution of nano-particle Figure 5. Fitting the two divisions of the nanoparticle’s intensity distribution with a Gaussian function. Figure 5. Fitting the two divisions of the nanoparticle’s intensity distribution with a Gaussian Whenfunction. the measured object plane is non-perpendicular to the optical axis, the images of a nanoparticle that move on this object plane can be considered to be a series of images on different object When the measured object plane is non-perpendicular to the optical axis, the images of a planes that are perpendicular to the optical axis (Figure6a). The nanoparticle is moved linearly on the nanoparticle that move on this object plane can be considered to be a series of images on different measured object plane to obtain a series of images of the linear position from one border position of the object planes that are perpendicular to the optical axis (Figure 6a). The nanoparticle is moved image plane to its opposite border position. For each image, the procedure shown in Figure5 was run, linearly on the measured object plane to obtain a series of images of the linear position from one and a series of differences between the two FWHMs was obtained for each image. Because the images’ border position of the image plane to its opposite border position. For each image, the procedure intensity distributions within the paraxial region had good symmetry, the differences between two shown in Figure 5 was run, and a series of differences between the two FWHMs was obtained for FWHMs within the paraxial region were smaller than those that were not. These differences generally each image. Because the images’ intensity distributions within the paraxial region had good presented a “U” shape, as shown as Figure6b, where the bottom of the line region is one line in the symmetry, the differences between two FWHMs within the paraxial region were smaller than those ideal imaging region of the measured object plane. Notably, the ideal imaging region of the measured that were not. These differences generally presented a “U” shape, as shown as Figure 6b, where the object plane (Figure6a) has some di fference from the ideal imaging region of Figure4 because this bottom of the line region is one line in the ideal imaging region of the measured object plane. series of images was obtained from different perpendicular object planes. In the ideal imaging region Notably, the ideal imaging region of the measured object plane (Figure 6a) has some difference from of the measured object plane, although isoplanatic conditions could not be met (the sizes of these the ideal imaging region of Figure 4 because this series of images was obtained from different spot images were not the same), the symmetries of the intensity distribution of these spot images still perpendicular object planes. In the ideal imaging region of the measured object plane, although performed well. isoplanatic conditions could not be met (the sizes of these spot images were not the same), the In order to confirm that one object plane is perpendicular to the optical axis, the two boundary symmetries of the intensity distribution of these spot images still performed well. points of the linear region must be obtained within the ideal imaging region of the measured object In order to confirm that one object plane is perpendicular to the optical axis, the two boundary plane. Here, a method of combining quartic polynomial curve fitting with logistic curve fitting was points of the linear region must be obtained within the ideal imaging region of the measured object used to confirm the boundary point (Figure6b). This process is called the polynomial combining plane. Here, a method of combining quartic polynomial curve fitting with logistic curve fitting was logistic curve fitting method and is described as follows. used to confirm the boundary point (Figure 6b). This process is called the polynomial combining In Figure6b, the black star point is a series of the di fferences between the two FWHMs. Fitting with logistic curve fitting method and is described as follows. a quartic polynomial curve to obtain the green line, the second vertex position is split into difference data, where the two divided parts are fitted with logistic curves. The fitting model is:

K L(s) = (2) 1 + R e U s · − · where K, R, and U are the undetermined coefficients that need to be confirmed through fitting; L is a dependent variable that is, in this case, the difference between the two FWHMs; and s is an Appl. Sci. 2020, 10, 2430 6 of 13 independent variable that is, in this case, the linear position when the nanoparticle is moved linearly on the measured object plane. The logistic curve fittings of the front and the rear division parts are represented by the red and blue lines, respectively (Figure6b). The position values of the red line’s second inflection point and the blue line’s first inflection point are (ln(Rred) – 1.317)/Ured and (ln(Rblue) – 1.317)/Ublue, respectively. The red line’s peak point and the blue line’s peak point are ln(Rred)/Ured and ln(Rblue)/Ublue, respectively. The line segment between the (ln(Rred) – 1.317)/Ured and (ln(Rblue)– 1.317)/Ublue points is chosen. The lowest of the difference values on the (2ln(Rred) – 1.317)/(2Ured) and (2ln(Rblue) – 1.317)/(2Ublue) positions is the threshold value. If the difference values of the positions within the chosen line segment are all below the threshold value, then the chosen line segment is the linear region within the ideal imaging region of the measured object plane, and two boundary points of the chosen line segment are determined. When two of the chosen line segments are obtained, then the four boundary points of the two chosen lines determine the ideal imaging region of the measured objectAppl. Sci. plane 2020, 1 (Figure, x FOR PEER6a). REVIEW 6 of 13

(a)

(b)

Figure 6. (a) Imaging of a nanoparticle moving on the measured object plane; (b) polynomial combining theFigure logistic 6. (a curve) Imaging fitting of method a nanoparticle to judge the moving ideal imagingon the regionmeasured of the object measured plane; object (b) polynomial plane. combining the logistic curve fitting method to judge the ideal imaging region of the measured object Toplane. obtain one object plane that is perpendicular to the optical axis and its paraxial region, the following steps should be followed: In Figure 6b, the black star point is a series of the differences between the two FWHMs. Fitting 1.with Keepa quartic the measuredpolynomial object curve plane’s to obtain space the attitude green forline, the the pitching second and vertex Z-axis position position is split constant. into differenceRoll the data, measured where the object two plane divided in aparts small are range fitted of with –Angle logisticto +Angle curves.. The fitting model is: 2. After each rolling operation, apply the polynomial combining logistic curve fitting method stated K above to determine the ideal imagingLs region()= of the−⋅ measured object plane. In the ideal imaging(2) 1+⋅ReUs region, choose the Gaussian function’s FWHM for one spot image as a reference object (here, the wherespot K, R, imaging and U are nearest the undetermined to the center ofcoefficients the ideal imagingthat need region to be confirmed was chosen). through Compare fitting; it L with is a dependentthe Gaussian variable function’s that is, FWHMin this case, for any the other difference function between in the two the chosen two FWHMs; lines within and the s is ideal an independentimaging variable region, obtain that is, the in dithisfference’s case, the mean linea squarer position root when (RMS) the for nanoparticle each compared is moved pair, record linearly the on the measured object plane. The logistic curve fittings of the front and the rear division parts are represented by the red and blue lines, respectively (Figure 6b). The position values of the red line’s second inflection point and the blue line’s first inflection point are (ln(Rred) –1.317)/Ured and (ln(Rblue)– 1.317)/Ublue, respectively. The red line’s peak point and the blue line’s peak point are ln(Rred)/Ured and ln(Rblue)/Ublue, respectively. The line segment between the (ln(Rred) –1.317)/Ured and (ln(Rblue)– 1.317)/Ublue points is chosen. The lowest of the difference values on the (2ln(Rred) –1.317)/(2Ured) and (2ln(Rblue) –1.317)/(2Ublue) positions is the threshold value. If the difference values of the positions within the chosen line segment are all below the threshold value, then the chosen line segment is the linear region within the ideal imaging region of the measured object plane, and two boundary points of the chosen line segment are determined. When two of the chosen line segments are obtained, then the four boundary points of the two chosen lines determine the ideal imaging region of the measured object plane (Figure 6a). To obtain one object plane that is perpendicular to the optical axis and its paraxial region, the following steps should be followed: 1. Keep the measured object plane’s space attitude for the pitching and Z-axis position constant. Roll the measured object plane in a small range of –Angle to +Angle. Appl. Sci. 2020, 10, 2430 7 of 13

average of these RMS values for the current rolling space attitude, and then choose the measured object plane corresponding to the smallest average of all rolling space attitudes. 3. Keep the rolling and Z-axis position of the obtained object plane space attitude constant. Pitch the object plane in a small range and carry out the same operations in steps 1–2. The chosen object plane on this step is one object plane that is perpendicular to the optical axis. Then, again run the polynomial combining logistic curve fitting method on the perpendicular object plane to confirm its ideal imaging region. The paraxial region on some perpendicular object plane has its corresponding ideal imaging region on the image plane (Figure4). Thus, the confirmation of this ideal imaging region is equal to the confirmation of the paraxial region of the perpendicular object plane.

3.3. D PSF Evaluation The fluorescent normalization intensity of one fluorescent nanoparticle on the object plane is [18]:  q q  2 2 2 2 2  1 (xo + yo )/(Dparticle/2) , ( xo + yo Dparticle/2) fparticle(xo, yo) =  − ≤ (3)  0, (otherwise) where xo and yo are the coordinate values of the object plane along the X and Y axes, respectively; Dparticle is the diameter of the fluorescent nanoparticle. When only considering geometrical optic imaging, the intensity distribution of the fluorescent nanoparticle image is fparticle (–xi/M,–yi/M) after microscopic magnification, where M is the microscopic magnification and xi and yi are the coordinate values of the imaging plane. The normalization intensity distribution of the PSF on an object plane can be considered a function of the Gaussian function shape [19,20]:

2 2 xi + yi I(x , y ) = exp( ) (4) i i − τ2 2 I where τI is standard deviation (τI is the desired value) and the PSF’s FWHM is determined by τI, FWHMPSF = 2τI √2ln2. The normalized microscopic diffraction imaging of one fluorescent nanoparticle is:

F (x , y ) = f ( x /M, y /M) I(x , y ) (5) imaging i i particle − i − i ∗ i i where Fimaging (xi, yi) is the normalized intensity distribution of the nanoparticle’s image and the Fimaging (xi, yi) central position can be obtained with the Gaussian function. In this way, it is possible to build the coordinates of Equation (5) to use as the central position for the original point. In order to obtain I (xi, yi) so that the standard deviation τI of I (xi, yi) can be obtained, Equation (5) can be used to solve a minimization problem:

Minimize : f ( x /M, y /M) I(x , y ) F (x , y ) (6) k particle − i − i ∗ i i − imaging i i k Equation (6) can be solved by an iteration algorithm. Choosing a fine initial iteration value plays a very important role in whether or not an ultimate optimum solution is obtained. Due to the FWHM of Fimaging (xi, yi), FWHMimaging is slightly greater than FWHMPSF; particularly, when Dparticle is small, the former is approximately equal to the latter. Therefore, the standard deviation τI of the initial iteration value can be taken as FWHMimaging/(2 √2ln2). 2 τIZ is the τI value corresponding to any object plane, and at the focal depth (τIZ) , the variation 2 3 2 approximately acts linearly [19]. Here, we used cubic polynomial (τIZ) = a ZP + b ZP + c ZP + 2 2 | | | | | | (τIo) to fit the (τIZ) variation, where ZP is any position relative to the focal object plane (original position) along the Z-axis; τIo is the τI value corresponding to the focal object plane; and a, b, and c are fitting coefficients. The 3D PSF can be expressed as: Appl. Sci. 2020, 10, 2430 8 of 13

2 + 2 xi yi IZ(x , y ) = exp( ) (7) i i − τ2 2 IZ

Obtaining τIZ on five perpendicular object planes (the focal object plane and any four perpendicular object planes) can initially provide the coefficients a, b, and c. Therefore, the initial iteration value τ q IZint 3 2 2 of Equation (6) can be taken as τIZ = a ZP + b ZP + c ZP + (τIo) for the process of obtaining | | | | | | τIZ on other perpendicular object planes. All obtained τIZ values and their corresponding ZP values are used to determine the new fitting coefficients, a, b, and c. This operation is repeated for the perpendicular object planes as much as possible, ultimately confirming the coefficients a, b, and c.

4. Entire 3D PSF Evaluation Procedure Regardless of the object plane’s perpendicularity, the ideal imaging region determination of the measured object plane (a polynomial combining the logistic curve fitting method), and the 3D PSF evaluation, it is necessary to measure the PSF calibration module’s spatial orientation and position through the dual PSD-based unit. Combining the statements in Section3, the total PSF evaluation procedure can be presented as the following steps:

1. Place a fluorescent nanoparticle under the microscope for imaging. Adjust the fiber-coupled laser to make its unit direction vector <0, 0, –1> in the coordinates determined by the dual PSD-based unit. 2. Determine the object plane’s perpendicularity and the paraxial region, as detailed in Section 3.2. There are a few points to note during this process: i. The PSF calibration module’s spatial orientation (rolling angle α and pitching angle β) and the PSF calibration module’s plane position both need to be confirmed. It is clear that the PSF calibration module’s plane position assurance can be realized by the dual PSD-based unit’s CCD (Figure2). The PSF calibration module’s spatial orientation can be confirmed by observing the variation of the PSF calibration module’s spatial orientation. The unit direction vector of the light beam from the fiber-coupled laser becomes , which is obtained by the dual PSD-based unit. Therefore, according to the transformation of the rectangular coordinates in space, α and β can be solved by the following matrix:    cos β 0 sin β   −  (mX, nY, qZ) = (0, 0, 1) sin α sin β cos α sin α cos β  (8) −    cos α sin β sin α cos α cos β  − ii. After each rolling (or pitching) operation in the procedure of obtaining one object plane that is perpendicular to the optical axis (Section 3.2), the measured object plane is the X–Y plane of the coordinates determined by the dual PSD-based unit, and the unwanted rolling angle and pitching angle caused by movement can also be obtained with Equation (8). The six-degree-freedom macro/micro displacement platform is able to compensate for unwanted spatial orientation according to the rolling and pitching angles that are needed to keep the PSF calibration module’s spatial orientation constant during the process of determining the ideal imaging region of the measured object plane (step 2 in Section 3.2). iii. After each rolling (or pitching) operation, the fiber-coupled laser must be adjusted to make its unit direction vector <0, 0, –1> again and to adjust the Z-position measurement CCD to make the light beam reflected by the semi-transparent mirror normal (namely, to make the unit direction vector of the light beam reflected by the semi-transparent mirror <0, 0, –1> in the coordinates determined by the dual PSD-based unit). Then, the PSF calibration module’s Z-axis direction position in its own coordinates can be confirmed (Z-axis direction position variation can be measured) to make sure the PSF calibration module (fluorescent particle) can stably move on the measured object plane (the X–Y plane of the dual PSD-based unit’s coordinates) and keep its longitudinal space position unchanged. Appl. Sci. 2020, 10, 2430 9 of 13

3. After the two steps above, one object plane that is perpendicular to the optical axis is confirmed, and the Z-axis of the dual PSD-based unit’s coordinates becomes the optical axis. The Z-position measurement CCD is adjusted to normally reflect the light beam by the semi-transparent mirror. The light is transmitted through the semi-transparent mirror incidents into the Z-position measurement CCD. Then, the PSF calibration module’s (fluorescent particle’s) Z-axis direction position (longitudinal space position along the optical axis) can be measured by the Z-position measurement’s CCD. 4. The polynomial combining logistic curve fitting method is run on different perpendicular object planes along the Z-axis (the optical axis) to determine the paraxial regions. This also determines the magnifications of different perpendicular object planes through the planar translational: magnification = the measured XY displacements on the image plane/the XY displacement measured by the dual PSD-based unit. 5. Finally, the 3D PSF evaluation in Section 3.3 is run for perpendicular object planes along the Z-axis to obtain the microscope’s 3D PSF.

5. Experiments and Results An experiment was used to verify the ability of the proposed method to obtain the 3D PSF. A standard 150 nm diameter fluorescent nanoparticle (Fluorescence Polystyrene, excitation peak: 540 nm; emission peak: 580 nm) was fixed in the cell of the proposed PSF measurement setup (Figure2). The objective lens’ nominal magnification was 60X (apochromatism objective 0.75 NA) using a commercial microscope, which needed to be calibrated. The entire 3D PSF evaluation procedure (Section4) was run to obtain the results of Figure7 and Table1. The focal object plane was used as the original position along the Z-axis, and it was demonstrated that the obtained PSF fitting line closely matched the original data (Figure7). The cubic polynomial coefficients for fitting the 3D PSF were obtained as detailed in Section 3.3 (Table1). For the consistency of presentation, here, the mentioned paraxial region was actually its corresponding ideal imaging region on the image plane. The intensity distribution of the nanoparticle’s image and its PSF Appl. Sci. 2020, 1, x FOR PEER REVIEW 10 of 13 intensity distribution on the original Z-axis position are shown in Figure8.

144.6 60.72 Fitting line 144.5 60.71 Orignal data

144.4 60.7

144.3 60.69

Magnification 144.2 (a) 60.68 (b)

Standard deviation of Standard PSF (nm) 144.1 60.67 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Z axis position (um) Z axis position (um)

2.3

2.2

2.1

2

1.9 (c) 1.8 Diameter of paraxial region (mm) of region Diameter paraxial -1 -0.5 0 0.5 1 Z axis position (um) Figure 7. The obtained 3D PSF (a), magnification (b), and paraxial region (c). Figure 7. The obtained 3D PSF (a), magnification (b), and paraxial region (c).

400 (b) 300

200

100

0

1.4 -100

1.2 -200

Position the on Y-axis (nm)

1 -300

0.8 run the 3D PSF -400 (a) evaluation in Section 3.3 -400 -300 -200 -100 0 100 200 300 400 0.6 Position on the X-axis (nm) 1 0.4 (c)

0.2 0.8

Intensity distribution of nano-particle 0 -100 100 -50 50 0.6 0 0 Position on the Y-axis (μm) 50 -50 Position on the X-axis (μm) 100 -100 0.4

Intensity distribution of PSF 0.2

0 -400 -300 -200 -100 0 100 200 300 400 Position on the X(Y)-axis (nm)

Figure 8. Intensity distribution of the nanoparticle’s image (a) and its PSF’s intensity distribution in the X–Y (b) and X–Z (or Y–Z) (c) planes.

In order to test the results of the 3D PSF evaluation, 100 and 200 nm diameter nanoparticles were measured on the three perpendicular object planes (the Z-axis positions of 0.5, 0, and -0.5 um) along the Z-axis (the optical axis of the commercial microscope), as well as the three obtained PSFs and magnifications corresponding to the three perpendicular object planes. We compared the FWHMs of the measured and calculated intensity distributions of the 100 and 200 nm diameter nanoparticles (the calculated intensity distributions could be determined by Equation (5)). The Appl. Sci. 2020, 1, x FOR PEER REVIEW 10 of 13

144.6 60.72 Fitting line 144.5 60.71 Orignal data

144.4 60.7

144.3 60.69 Magnification 144.2 (a) 60.68 (b)

Standard deviation of PSF (nm) 144.1 60.67 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Z axis position (um) Z axis position (um)

2.3

2.2

2.1 Appl. Sci. 2020, 10, 2430 10 of 13 2

1.9 Table 1. Cubic polynomial coefficients of fitting(c) ZP and τIZ. 1.8 Diameter of paraxial region (mm) -1a-0.5 0 b0.5 1 c τ2 Z axis position (um) Io 30 30 16 14 Z-axis positive direction 8.11 10− 9.86 10− 1.11 10− 2.08 10− × 28 − × 28 − × 16 × 14 Z-axis negativeFigure direction 7. The obtained1.14 3D10 PSF− (a), magnification1.64 10− (b), and1.12 paraxial10− region (c2.08). 10− × − × × ×

400 (b) 300

200

100

0

1.4 -100

1.2 -200 Position on the Y-axis (nm) Y-axis the on Position

1 -300

0.8 run the 3D PSF -400 (a) evaluation in Section 3.3 -400 -300 -200 -100 0 100 200 300 400 0.6 Position on the X-axis (nm) 1 0.4 (c) 0.2 0.8

Intensity distribution of nano-particle 0 -100 100 -50 50 0.6 0 0 Position on the Y-axis (μm) 50 -50 Position on the X-axis (μm) 100 -100 0.4

Intensity distribution of 0.2PSF

0 -400 -300 -200 -100 0 100 200 300 400 Position on the X(Y)-axis (nm) Figure 8. a Figure 8. IntensityIntensity distributiondistribution of of the the nanoparticle’s nanoparticle’s image image ( )( anda) and its PSF’sits PSF’s intensity intensity distribution distribution in the in X–Y (b) and X–Z (or Y–Z) (c) planes. the X–Y (b) and X–Z (or Y–Z) (c) planes. In order to test the results of the 3D PSF evaluation, 100 and 200 nm diameter nanoparticles were In order to test the results of the 3D PSF evaluation, 100 and 200 nm diameter nanoparticles measured on the three perpendicular object planes (the Z-axis positions of 0.5, 0, and –0.5 µm) along were measured on the three perpendicular object planes (the Z-axis positions of 0.5, 0, and -0.5 um) the Z-axis (the optical axis of the commercial microscope), as well as the three obtained PSFs and along the Z-axis (the optical axis of the commercial microscope), as well as the three obtained PSFs magnifications corresponding to the three perpendicular object planes. We compared the FWHMs and magnifications corresponding to the three perpendicular object planes. We compared the of the measured and calculated intensity distributions of the 100 and 200 nm diameter nanoparticles FWHMs of the measured and calculated intensity distributions of the 100 and 200 nm diameter (the calculated intensity distributions could be determined by Equation (5)). The FWHM differences nanoparticles (the calculated intensity distributions could be determined by Equation (5)). The between the measured FWHMs and the calculated FWHMs on the three perpendicular object planes demonstrated that the differences clearly defined the paraxial region (Figure9). The plane position was the distance from the center of the paraxial region. The FWHM differences in the paraxial region (the red line) were almost all below 5 nm (Figure9). The FWHM di fferences became bigger with an increase in the deviation from the paraxial region (the black line; Figure9). This verified that the obtained 3D PSF, magnification, and paraxial region (Figure7) were correct and also demonstrated the validity of the proposed PSF evaluation method to calibrate the object plane that was perpendicular to the optical axis. Appl. Sci. 2020, 1, x FOR PEER REVIEW 11 of 13

FWHM differences between the measured FWHMs and the calculated FWHMs on the three perpendicular object planes demonstrated that the differences clearly defined the paraxial region (Figure 9). The plane position was the distance from the center of the paraxial region. The FWHM differences in the paraxial region (the red line) were almost all below 5 nm (Figure 9). The FWHM differences became bigger with an increase in the deviation from the paraxial region (the black line; Figure 9). This verified that the obtained 3D PSF, magnification, and paraxial region (Figure 7) were correct and also demonstrated the validity of the proposed PSF evaluation method to calibrate the Appl. Sci. 2020, 10, 2430 11 of 13 object plane that was perpendicular to the optical axis.

40 50 Non-paraxial region Non-paraxial region 35 Paraxial region Paraxial region 40 30

25 30 20

15 20

10 10 5 (a) (b) Absolute value of difference (nm) FWHM 0 Absolute value of difference (nm) FWHM 0 -2 -1 0 1 2 -2 -1 0 1 2 Plane position (mm) Plane position (mm)

50 60 Non-paraxial region Non-paraxial region Paraxial region 50 Paraxial region 40

40 30 30 20 20

10 10 (c) (d) Absolute value of difference (nm) FWHM 0 Absolute value of difference (nm) FWHM 0 -2 -1 0 1 2 -2 -1 0 1 2 Plane position (mm) Plane position (mm) 40 50 Non-paraxial region Non-paraxial region 35 Paraxial region Paraxial region 40 30

25 30 20

15 20

10 10 5 (e) (f) Absolute value of difference (nm) FWHM 0 Absolute value of difference (nm) FWHM 0 -2 -1 0 1 2 -2 -1 0 1 2 Plane position (mm) Plane position (mm) Figure 9. Full widths at half maximum (FWHM) differences between the measured FWHM and the Figurecalculated 9. Full FWHM widths for at standard half maximum 100 nm ( a(,FWHMc,e) and) 200 differences nm (b,d,f )between diameter the nanoparticles: measured FWHM On the 0.5 andµm the calculatedZ-axis position FWHM (a ,forb); onstandard the original 100 nm Z-axis (a,c position,e) and 200 (c,d );nm and (b on,d,f the) diameter –0.5 µm Z-axisnanoparticles: position (One,f). the 0.5 um Z-axis position (a,b); on the original Z-axis position (c,d); and on the -0.5 um Z-axis position (e,f). 6. Conclusions 6. ConclusionIn this paper,s a point spread function (PSF) evaluation method for a microscope on the object planeIn this that paper, is perpendicular a point spread to the function optical ( axisPSF) was evaluation proposed. method This for method a microscope mainly includes on the object the planemeasurement that is perpendicular of a dual PSD-based to the unit’s optical incident axis was beam proposed. direction, This the determination method mainly of the includes paraxial the measurementregion and the of objecta dual plane’s PSD-based perpendicularity, unit’s incident and beam the evaluation direction, of the the determination 3D PSF. The PSF of calibrationthe paraxial module involves a dual PSD-based unit that is used to achieve the spatial orientation and position location of the object plane and the fluorescent particle over the entire PSF evaluation process. The proposed method determines the magnification and paraxial region on each perpendicular object plane in the produced 3D PSF evaluation. A standard 150 nm nanoparticle was employed in the proposed PSF evaluation method to obtain the 3D PSF, the magnification, and the paraxial region of Appl. Sci. 2020, 10, 2430 12 of 13 the microscope. A comparison of the 100 and 200 nm nanoparticles’ measured FWHMs and calculated FWHMs demonstrated the validity of the proposed PSF evaluation method to obtain the 3D PSF, the magnification, and the paraxial measurements.

Author Contributions: S.M. conceived the experiments, designed the experiments, performed the experiments, analyzed the data, and wrote the paper; Z.W. analyzed the data and designed the experiments; J.P. conceived the experiments. All authors have read and agreed to the published version of the manuscript. Funding: Research reported in this study is supported by China Postdoctoral Science Foundation (2019M652440), Shandong Provincial Natural Science Foundation, China (Grant No. ZR2018PF014, Grant No. ZR2018MF032, Grant No. ZR2017LF026) and the National Natural Science Foundation of China (Grant No. 61801272). Conflicts of Interest: The authors declare no conflict of interest.

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