Microshape Measurement Method Using Speckle Interferometry Based on Phase Analysis

hv photonics

Article Microshape Measurement Method Using Speckle Interferometry Based on Phase Analysis

Yasuhiko Arai

Department of Mechanical Engineering, Faculty of Engineering Science, Kansai University, Yamate-cho 3-3-35, Japan; [email protected]

Abstract: A method for the measurement of the shape of a ﬁne structure beyond the diffraction limit based on speckle interferometry has been reported. In this paper, the mechanism for measuring the shape of the ﬁne structure in speckle interferometry using scattered light as the illumination light is discussed. Furthermore, by analyzing the phase distribution of the scattered light from the surface of the measured object, this method can be used to measure the shapes of periodic structures and single silica microspheres beyond the diffraction limit.

Keywords: measurement beyond the diffraction limit; measurement mechanism; non-periodical ﬁne structure measurement; speckle interferometry; detecting phase distribution

1. Introduction When a rough surface is illuminated with a highly coherent light, scattered light is generated from the measured surface. The scattered lights interfere with each other and produce speckle, which is a grain interference fringe whose diameter depends on the size of the aperture of the objective in the optical system [1–3]. In interference measurements, speckle is sometimes treated as noise present in fringe Citation: Arai, Y. Microshape images. Therefore, a reduction in the influence of speckle noise is important in speckle inter- Measurement Method Using Speckle ference measurement technology [4,5]. Filtering technology has been studied and further Interferometry Based on Phase developed to address speckle noise [5–7]. In addition, high-resolution measurements using Analysis. Photonics 2021, 8, 112. techniques such as fringe scanning [3] have been widely employed in speckle interferome- https://doi.org/10.3390/ try. Several high-resolution measurement methods for deformation or displacement have photonics8040112 been developed by analyzing the recorded phase distribution in the speckle [8–13]. Speckle is currently considered a significant phenomenon in the optical field. Furthermore, speckle Received: 21 February 2021 interferometry has been used for three-dimensional (3D) shape measurements [14,15]. A Accepted: 6 April 2021 Published: 8 April 2021 3D structure beyond the diffraction limit of the objective lens can be measured by using the scattered light as the illumination light and by detecting the phase of the light recorded

Publisher’s Note: MDPI stays neutral in the speckle formed by the scattered light [15]. with regard to jurisdictional claims in The principle of this method [14,15] can be validated by comparing its measurement published maps and institutional afﬁl- results with those of other techniques, such as atomic force microscopy (AFM), using iations. the grating of the periodic structure as the measurement object. In this case, because the same structure exists everywhere, the same situation of the same measured object can be measured [14–16]. However, if only the periodic structure is measured, misunderstandings and misconceptions can emerge if the speckle interference measurement technology is not well understood. It has been misunderstood that the proposed method can be used Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. only for measuring the periodic structure. In addition, a misconception emerged that this This article is an open access article method has the same principle as that of scatterometry [17–20], which is used to measure distributed under the terms and periodic structures. conditions of the Creative Commons To clarify such misunderstandings and misconceptions, this study demonstrates that Attribution (CC BY) license (https:// the proposed measurement method can measure three-dimensional structures that exist creativecommons.org/licenses/by/ randomly in space. Furthermore, the same measured object at the same sample position 4.0/). was measured under different conditions of the diffraction limit by installing an aperture

Photonics 2021, 8, 112. https://doi.org/10.3390/photonics8040112 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 112 2 of 13

in front of the objective lens. Accordingly, the results were analyzed based on whether the diffraction limit exceeded. As a result of comparing the results under the two situations, it was conﬁrmed that the proposed method can measure beyond the diffraction limit. Furthermore, this paper explains the mechanism of the measurement process based on the principle of this measurement method using scattered light as illumination. This paper provides an understanding of the processing of the method, which makes it possible to perform three-dimensional measurements by analyzing only the phase change of the scattered light from the object without using the higher-order diffracted light based on conventional imaging theory [21].

2. Method In a conventional speckle interferometer [1], the laser beam was divided by half mirrors. One of the obtained beams was irradiated onto the measurement object with a rough surface. The other was irradiated onto a rough reference surface to produce the reference light. An interference is generated between the scattered lights from the rough surfaces. The interference fringes, displayed on the image sensor, are captured through lens [1,11–13]. When such an optical system is used, the sampled speckle pattern has a bias component and original signal component in the low-frequency region in the intensity distribution. This phenomenon is the same as the interference fringe intensity distribution for a general interferometer because speckle is also an interference phenomenon. Therefore, an image in which the significant signal component in the frequency domain superimposes on the bias is obtained [6]. Such an image with a high resolution can be processed and a high phase component can be extracted using temporal fringe scanning technique [1–3]. However, this analysis technique cannot be used for dynamic analysis because three sheets of speckle patterns are required. In addition, because each speckle before and after the deformation must overlap in the speckle interference measurement [1,22,23], the relationship between the deformation and speckle size poses a problem in large- deformation measurements when the three speckle patterns are captured [24]. However, when a plane wave is used as the reference beam and the wave plane is angled with respect to the object beam [5], a carrier signal can be provided in the speckle. By providing such a carrier signal and performing a Fourier transform on the captured speckle pattern, the bias and signal components can be separated in the frequency domain [6,7,23–26]. The bias component can be removed by extracting only the signal components in this process. Thus, the phase component can be obtained using only two speckle patterns before and after deformation [22,23]. Furthermore, the phase distribution of the specklegram between images was analyzed using the basic speckle interferometry [5]. The noise in the low-frequency region of these images was removed by a filter using the Fourier transform. However, when lasers were used, speckles in images were historically treated as noise components, and methods to reduce speckle noise have been extensively studied. In the analysis method used in this study, the noise components in the speckle are reduced by a two-step filtering technique to ensure that they do not affect the analysis. Through this processing [7], phase analysis is achieved without affecting the signal components. Using such filtering technologies, the phase information in speckle interferometry can be extracted smoothly by a camera and computer technology, such as electronic speckle pattern interferometry (ESPI) [1,2]. Deformation measurements with a resolution of several tens of nanometers were conducted using such technologies [22,23]. On the contrary, in terms of the fine structure used in this study, it is believed that it is impossible to obtain the final shape unless filtering is performed. In this study, as described above, processing using filtering technology for noise components in two different frequency bands is performed in two steps [14–16]. Therefore, it is possible to observe the fine structure smoothly. Photonics 2021, 8, 112 3 of 13

The cross-section of the measured object is shown in Figure1. In Figure1a, the phase cross-section of the speckle pattern is shown when the measured object is set to the original position of the speckle interferometer. In Figure1b, the phase cross-section of the speckle pattern is set at the time when the measured object is shifted by a small amount (δx). When a specklegram is calculated between these two speckle patterns, the phase change at each position can be analyzed with high resolution using speckle interferometry. Assuming that the phase distribution related to the shape of the measured object is f(x), as shown in Figure1a, the phase change caused by the lateral shift ( δx) at Pa in the original position is the difference between Pa and Pb, i.e., f(x) − f(x + δx). If this phase change is divided by the shift (δx), the pseudo-differential value (∂f/∂x) of the phase distribution f(x) related to the shape can be obtained, as shown in Figure1. By integrating this differential value

with respect to the shifted coordinates, the phase distribution f(x) related to the shape of the original object can be obtained.

Figure 1. Principal of analyzing method: (a) Section of measured object at original position; (b) Section of measured object at shifted position.

As mentioned earlier, the lateral shift of the measured object is believed to be an “advantageous deformation for the speckle deformation measurement” for measurements of 3D shapes based on speckle interferometry. In addition, a piezo element was used in the initial experiment set for achieving lateral shift [14]. However, as the shift is several tens of nanometers, although the speckle pattern data collected without horizontal shifting are virtually shifted in the computer, it was reported that the result of the virtual shift produces the same effect as the result of the shift using the piezo element [15]. Therefore, in this analysis, only one speckle pattern was captured instead of collecting two speckle patterns before and after the horizontal shift. A second speckle pattern is produced as image data by virtually shifting in the memory of the computer [16]. Thus, the 3D shape can be measured using only one speckle pattern without any influence, such as mechanical vibration and fluctuation in air, from the experimental environment. In the 3D shape measurement, when a plane wave is used as the illumination light in the speckle interferometer, the scattered light from the measured object surface is largely affected by diffraction when the structure of the measurement surface becomes finer. Thus, the distribution of the diffracted light increases. When the higher-order diffracted light cannot pass through the lens area, the amount of scattered light passing through the lens from the surface of the measured object decreases. However, the decrease in the

Forests 2021, 12, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/forests PhotonicsPhotonics 20212021, 8,, x8 ,FOR 112 PEER REVIEW 4 of4 of14 13

in amountthe amount of scattered of scattered light light poses poses a problem a problem in in this this method, method, based based on on the the phase phase analysis anal- ysisof of scattered scattered light. light. In In suchsuch aa case,case, it is advantageous advantageous to to use use scattered scattered light light with with many many lightlight direction direction vectors vectors as asillumination illumination light light to toobtain obtain more more scattered scattered light light through through the the lens. lens. Accordingly,Accordingly, scattered scattered light light was was used used as as the the illumination illumination light light at atthe the time time of ofmeasurement measurement in inthis this study. study. TheThe speckle speckle interferometer interferometer used used in in this this experiment experiment is is shown shown in in Figure Figure 22.. In thisthis opticalopti- calsystem, system, scattered scattered light light generated generated after after passing passing through through the the ground ground glass glass was was used used as as the theillumination illumination light. light. The The reﬂected reflected scattered scattered light light from from the the surface surface of theof the object objec wast was collected col- lectedby an by objective an objective lens (productlens (product number: number: M Plan M Plan Apo200 Apo200×,×, NA: NA: 0.62, 0.62, magniﬁcation: magnification: 200× , 200×Mitutoyo, Mitutoyo Co. Co. Ltd.: Ltd. Kawasaki,: Kawasaki, Japan). Japan The). The scattered scattered light, light, including including the phasethe phase information, information,reaches reaches the image the sensor image (pixel sensor size: (pixel 1.6 size:µm, number 1.6 µm, of number pixels: 1024of pixels:× 1024) 1024 through × 1024) the throughpinhole, the as pinhole, shown inas Figureshown2 .in Figure 2.

Camera Mirror-1 BS

Pin hole Reference Lens 1/2 wave plate beam Collimator PBS Laser Micrometer Objective Collimator 1/2 wave plate Aperture BS Ground Sample glass

FigureFigure 2. Optical 2. Optical system. system.

TheThe beam beam emitted emitted from from the the same same laser laser (wavelength: (wavelength: 532 532 nm, nm, 150 150 mW) mW) is issplit split by by a a polarizingpolarizing beam beam splitter, splitter, collimated collimated by by a collimator, a collimator, and and used used as asthe the reference reference light. light. The The cameracamera captures captures the the speckle speckle pattern pattern originating originating fromfrom thethe interferenceinterference between between the the scattered scat- teredlight light from from the measuredthe measured object object and theand reference the reference light. light. The carrierThe carrier fringes fringes in the in speckle the speckleare generated are generated with thewith object the object light fromlight thefrom object the object surface surface and reference and reference beam beam by setting by settingthe angle the angle of mirror-1 of mirror to remove-1 to remove the bias the component bias component of the speckle of the pattern.speckle pattern. The frequency The frequencyof the carrier of the fringes carrier is fringes set considering is set considering the speckle the diameter, speckle diameter, transmission transmission characteristics char- of acteristicsthe optical of system,the optical and system, camera and characteristics camera characteristics shown in a previous shown i reportn a previous [24]. report [24]. Furthermore, in this optical system, an aperture with a diameter of 8 mm on the rail andFurthermore, a thickness ofin 0.5this mm optical was system, installed an in aperture front of with the objective a diameter lens of using 8 mm a on micrometer. the rail andThe a thickness NA of the of lens 0.5 ismm the was original installed value in offront 0.62, of when the objective the aperture lens using is not a inserted. micrometer. When Thethe NA aperture of the lens was is inserted, the original the aperturevalue of diameter0.62, when of the the aperture lens changed is not frominserted. 13 to When 8 mm, theresulting aperture in was an inserted, NA of 0.29. the Usingaperture such diameter an aperture, of the thelens diffraction changed from limit 13 of to the 8 opticalmm, resultingsystem in can an be NA changed of 0.29. without Using such involving an aperture, the measured the diffraction object orlimit the of construction the optical sys- of the temoptical can be system changed [16]. without In other involving words, the the optical measured system object can be or usedthe construction under two measurement of the op- ticalconditions, system [16]. except In other for thewords, introduction the optical of system apertures. can be Therefore, used under an two object measurement on the order of several hundred nanometers can be observed by exceeding and not exceeding the conditions, except for the introduction of apertures. Therefore, an object on the order of diffraction limit. In this optical system, this can be achieved by changing the diffraction several hundred nanometers can be observed by exceeding and not exceeding the diffrac- limit at the same measurement position of the same camera. The measurement results tion limit. In this optical system, this can be achieved by changing the diffraction limit at under different conditions are compared. the same measurement position of the same camera. The measurement results under dif- It is usually challenging to measure the microstructure (on the order of hundreds ferent conditions are compared. of nanometers) of the measured object at the same location using different measuring instruments. Therefore, in previous studies [14–16], the principle was validated by over-

Photonics 2021, 8, x FOR PEER REVIEW 5 of 14

It is usually challenging to measure the microstructure (on the order of hundreds of nanometers) of the measured object at the same location using different measuring instru- Itments. is usually Therefore, challenging in previous to measure studies the [14 microstructure–16], the principle (on thewas order validated of hundreds by overcoming of Photonics 2021, 8, 112 5 of 13 nanometers)this problem of the usingmeasured a diffraction object at thegrating same with location a periodic using differentstructure measuring by comparing instru- it with ments.other Therefore, measurement in previous techniques, studies [14such–16], as theAFM. principle This study was validatedverifies that by overcomingthis method can this problemmeasure usingnot only a diffraction the shapes grating of periodic with structuresa periodic butstructure also the by shapes comparing of randomly it with dis- othertributed measurementcoming 3D this objects. problem techniques, using such a diffraction as AFM. gratingThis study with verifies a periodic that structurethis method by comparing can measureit withnotIn thisonly other study, the measurement shapes the shape of periodic techniques,of the microstructure structures such as but AFM. randomlyalso Thisthe shapes study distributed veriﬁesof randomly in that 3D thiswas dis- methodmeas- tributedured.can 3D measureThe objects. measured not only objects the shapeswere silica of periodic spheres, structures whose shapes but alsowere the not shapes periodic. of randomlyIn addi- Intion,distributed this the study, diameter 3Dthe objects.shape of the of the spheres microstructure was standardized. randomly H distributedere, because in it3D is was challenging meas- to ured.measure The measuredIn thisthe same study, objects position the were shape of silica a of random the spheres, microstructure object whose using shapes different randomly were measuring not distributed periodic. instruments, in In 3D addi- was mea-the tion, measurement thesured. diameter The measured ofmethod the spheres was objects validated was were standardized. by silica comparing spheres, Here, the whose becausemeasured shapes it diameters is were challenging not of periodic. the to silica In measuresphere,addition, the samewhich the position has diameter a standardized of ofa random the spheres diameter object was using standardized.under different different measuring Here, measurement because instruments, it conditions. is challenging the The to measurementdiametermeasure methodof the the same silica was position spherevalidated ofwas a by random measured comparing object under the using measureddifferent different conditions diameters measuring ofof instruments,thethe diffractionsilica the sphere,limitmeasurement which without has achanging standardized method the was position diameter validated of underthe by comparingobject. different themeasurement measured diametersconditions. of The the silica diametersphere, ofThe the silica which silica spheres hassphere a standardized used was in measured this experiment diameter under under differentwere differentKMP conditions-590 measurement and X of-52 the-854 diffraction conditions.manufactured The limit bywithoutdiameter Shin- Etsuchanging of Chemical the silica the position sphere(Tokyo, wasof Japan) the measured object.. The former under is different a sphere conditions with a standard of the diffractiondiameter Theoflimit 2 silicaµm without (diameter spheres changing used distribution: in thethis positionexperiment 1–4 µm), of the werewhile object. KMP the -latter590 and is aX sphere-52-854 withmanufac a standardtured di- by Shinameter-EtsuThe of Chemical 700 silica nm spheres (diameter(Tokyo, used Japan) distribution: in this. The experiment former 0.2–5 isµm). were a sphere As KMP-590 shown with in anda standardFigure X-52-854 3, diameterth manufacturede powdered of 2 µmsampleby (diameter Shin-Etsu microsphere distribution: Chemical was (Tokyo, fixed 1–4 toµm), Japan). a 10 while mm The ×the former5 mmlatter test isis a piecea sphere sphere with with with an a adhesivea standard standard to diameter di-prepare of ameterthe2 of µsample m700 (diameter nm for (diameter measurement. distribution: distribution: In 1–4 addition,µm), 0.2– while5 µm).the thesurface As latter shown of is the a in sphere objectFigure with was 3, th acoatede standard powdered with diameter plati- samplenumof microsphere 700 with nm a (diameterthickness was fixed of distribution: several to a 10 nanometers mm 0.2–5 × 5 µmmm). by test As sputtering. shownpiece with in Figure an adhesive3, the powdered to prepare sample the samplemicrosphere for measurement. was ﬁxed In to addition, a 10 mm the× 5surface mm test of the piece object with was an adhesivecoated with to plati- prepare the num withsample a thickness for measurement. of several nanometers In addition, by the sputtering. surface of the object was coated with platinum with a thickness of several nanometers by sputtering.

(a) (b) 10 mm (a) (b) Figure 3. Test pieces with the10 microspheres mm (Silica sphere), (a) Diameter: 2.0 μm, (b) Diameter: 700 nm. Figure 3.Figure Test pieces 3. Test with pieces the with microspheres the microspheres (Silica (Silicasphere) sphere),, (a) Diameter: (a) Diameter: 2.0 μ 2.0m, µ(bm,) Diameter: (b) Diameter: 700 700 nm. nm. The diameter distribution of the measured object was observed using scanning electron microscopyThe diameter (SEM) distribution before performing of the measured the measurements. object was observed The usingresults scanning are shown electron in TheFiguremicroscopy diameter 4. (SEM)distribution before of performing the measured the measurements.object was observed The results using arescanning shown elec- in Figure4. tron microscopy (SEM) before performing the measurements. The results are shown in Figure 4.

2.5 μm 1.0 μm

2.5(a) μm 1.0(b) μm

FigureFigure(a) 4. 4.SEMSEM image image of of measured measured(b) objects:objects: (a ()a Diameter:) Diameter: 2.0 2.0µm μ (KMP-590);m (KMP-590); (b) Diameter: (b) Diameter: 700 nm 700 (X-52-854). nm (X-52-854). Figure 4. SEMThe image SEM of image measured shows objects: that ( thea) Diameter: average diameter2.0 μm (KMP of the-590); KMP-590 (b) Diameter: spheres 700 was nm approxi- (X-52-854).mately The 2SEMµm, image while thatshows of X-52-854 that the wasaverage approximately diameter of 700 the nm. KMP The-590 measured spheres diameter was ap- of proximatelythe sphere was2 µm, consistent while that with of thatX-52 in-854 the was ofﬁcial approximately product catalog. 700 nm. Although The measured the measured di- Theameterobject SEM of is imagethe approximately sphere shows was that consistent spherical, the average itwith is not diameterthat necessarily in the of official the an KMP idealproduct-590 sphere; spherescatalog. some Althoughwas ellipsoids ap- the as proximatelymeasureddistorted 2 µm, object spheres while is areapproximately that observed. of X-52-854 spherical, was approximately it is not necessarily 700 nm. Thean idealmeasured sphere; di- some ameterellipsoids of the sphere as distorted was consistent spheres withare observed. that in the official product catalog. Although the measured3. Results object is approximately spherical, it is not necessarily an ideal sphere; some ellipsoids3.1. as Shape distorted Measurement spheres of are Randomly observed. Distributed 3D Objects Not Exceeding the Diffraction Limit During the observation of the measured objects with diameters of 2 µm using the optical system shown in Figure2, a speckle pattern within a region with dimensions of approximately 20 µm × 20 µm was captured, as shown in Figure5a.

Photonics 2021, 8, x FOR PEER REVIEW 6 of 14

3. Results 3.1. Shape Measurement of Randomly Distributed 3D Objects Not Exceeding the Diffraction Limit During the observation of the measured objects with diameters of 2 µm using the Photonics 2021, 8, 112 6 of 13 optical system shown in Figure 2, a speckle pattern within a region with dimensions of approximately 20 µm × 20 µm was captured, as shown in Figure 5a.

FigureFigure 5. Speckle 5. Speckle pattern pattern and andmeasured measured result: result: (a) Measurement (a) Measurement area areain the in speckle the speckle interferogram; interferogram; (b) Area-A;(b) Area-A; (c) Three-dimensional (c) Three-dimensional shape. shape.

TheThe bright bright parts parts were were observed observed in the in thespeckl specklee pattern. pattern. In speckle In speckle interferometry, interferometry, the the measuredmeasured objects objects are areonly only the thesilica silica spheres spheres distributed distributed before before and and after after the theconjugate conjugate pointpoint of the of image the image formation formation position position of the of lens. the The lens. field The depth field of depth the objective of the objective lens used lens in thisused study in this was study ±692 was nm± because692 nm becausethe NA was the NA0.62, was and 0.62, the andlaser the wavelength laser wavelength was 532 was nm.532 The nm. value The of value the offield the depth field depthof the of obje thective objective lens lenswas wascalculated calculated using using the theformula formula λ 2 λ/(2NA/(2NA2), which), which was wasintroduced introduced by its by manufacturer, its manufacturer, i.e., i.e.,only only the thesilica silica spheres spheres existing existing ± in thein therange range of ±692 of 692nm nmcan canbe measured be measured during during the themeasurement measurement of the of therandomly randomly and and spatiallyspatially distributed distributed silica silica spheres spheres using using this method. this method. Therefore, Therefore, the phase the phase distribution distribution in Area-A,in Area-A, which which is a bright is a brightspot as spot a speckle, as a speckle, was measured was measured because because it was itassumed was assumed to be to be the measured area. An enlarged view of Area-A is shown in Figure5b. Some speckles the measured area. An enlarged view of Area-A is shown in Figure 5b. Some speckles were scattered, and carrier fringes were observed. Carrier fringes are used to remove the were scattered, and carrier fringes were observed. Carrier fringes are used to remove the bias component of the speckle pattern. Photonics 2021, 8, x FOR PEER REVIEWbias component of the speckle pattern. 7 of 14 The high-resolution phase distribution of Area-A (2.5 µm × 3 µm) of the speckle The high-resolution phase distribution of Area-A (2.5 µm × 3 µm) of the speckle pat- pattern shown in Figure5a, which is the bright area of the speckle pattern, was calculated tern shown in Figure 5a, which is the bright area of the speckle pattern, was calculated by by the principle of this method as a three-dimensional shape map, as shown in Figure5c. the principle of this method as a three-dimensional shape map, as shown in Figure 5c. directionIn this from operation, such positions though thiswere process simply has arranged been discussed as the two in the-dimensional previous papers result [in14 ,the15] In this operation, though this process has been discussed in the previous papers yin-direction. detail, the analyzing process is briefly introduced with the flowchart shown in Figure6. [14,15] in detail, the analyzing process is briefly introduced with the flowchart shown in

Figure 6.

In this process, first, one speckle pattern (S1) is captured. The acquired image is

shifted by a lateral shift to generate the second speckle pattern (S2) [15]. From these

speckle patterns, a process using the Fourier transform is performed to obtain the speck-

legram [6,7,24]. A high-resolution phase map corresponding to the derivative of the shape

of the measurement object is extracted by using the spatial fringe analysis method under

the idea of phase difference method [6,27]. The shape of the measurement object is then

obtained by integrating the derivative distribution in the direction of lateral shift [14,15].

In the two-dimensional calculation of the shape, the current operation is only shifting

in the x-direction, and there is no information in the y-direction. Therefore, the operation

in the y-direction was performed by using the idea in previous paper [28]. In the y-direc-

tion operation, the initial value of the x-direction operation at the position away from the

measurement object where the derivative distribution of the shape in the y-direction is

considered not to exist can be assumed zero. Then, the results of integration in each x-

FigureFigure 6. 6. FlowchartFlowchart of of processing processing in in Figure Figuress 5 and7 7..

Two peaks (vertices), P1 and P2, with different heights, were observed in the 3D shape map in Figure 5c. Furthermore, Figure 7a presents the contour map of Figure 5c with a contour width of 50 nm. The difference in height was also observed when these measurement results were displayed on a contour map, as shown in Figure 7a.

:Measured surface

1.5 :Curve fitting result m]

 (0.14, 0.36) 0 C1 0 x [m] 1.0 1.0 Height [ (b)

:Measured surface

1.5 :Curve fitting result m]



2(0.03, 0.25) 0 C 0 x [m] 1.0

Height Height [ 1.0 (c) (a)

Figure 7. Measured results: (a) Contour of 3D shape; (b) Measured shape of section A-A; (c) Meas- ured shape of section B-B.

Although the measured silica object includes some irregular spheres, most of the objects are approximately spherical, as shown in Figure 4. The vertex of the silica sphere can be easily estimated by observing the contour map shown in Figure 7a. Considering that the measured object is a sphere, it can be assumed that the horizontal cross-section, including the vertex on the contour map, is the cross-section of the sphere. Thus, the A–A and B–B cross-sections that include P1 and P2 can be set parallel to each x-coordinate. Therefore, the cross-section of the sphere can be easily obtained. Furthermore, as shown

Photonics 2021, 8, 112 Forests 2021, 12, x FOR PEER REVIEW 2 of 3 7 of 13

Figure 7. Measured results: (a) Contour of 3D shape; (b) Measured shape of section A-A; (c) Measured shape of section B-B.

In this process, first, one speckle pattern (S1) is captured. The acquired image is shifted by a lateral shift to generate the second speckle pattern (S2) [15]. From these speckle patterns, a process using the Fourier transform is performed to obtain the specklegram [6,7,24]. A high-resolution phase map corresponding to the derivative of the shape of the measurement object is extracted by using the spatial fringe analysis method under the idea of phase difference method [6,27]. The shape of the measurement object is then obtained by integrating the derivative distribution in the direction of lateral shift [14,15]. In the two-dimensional calculation of the shape, the current operation is only shifting in the x-direction, and there is no information in the y-direction. Therefore, the operation in the y-direction was performed by using the idea in previous paper [28]. In the y-direction operation, the initial value of the x-direction operation at the position away from the measurement object where the derivative distribution of the shape in the y-direction is considered not to exist can be assumed zero. Then, the results of integration in each x-direction from such positions were simply arranged as the two-dimensional result in the y-direction. Two peaks (vertices), P1 and P2, with different heights, were observed in the 3D shape map in Figure5c. Furthermore, Figure7a presents the contour map of Figure5c with a contour width of 50 nm. The difference in height was also observed when these measurement results were displayed on a contour map, as shown in Figure7a. Although the measured silica object includes some irregular spheres, most of the objects are approximately spherical, as shown in Figure4. The vertex of the silica sphere can be easily estimated by observing the contour map shown in Figure7a. Considering that the measured object is a sphere, it can be assumed that the horizontal cross-section, including the vertex on the contour map, is the cross-section of the sphere. Thus, the A–A and B–B cross-sections that include P1 and P2 can be set parallel to each x-coordinate. Therefore, the cross-section of the sphere can be easily obtained. Furthermore, as shown in Figure4, a 3D estimation method based on the exact calculation of the diameter was not selected because the width of the diameter distribution is between 1 and 4 µm and silica objects are not necessarily ideal spheres. The measurement results by two-dimensional curve (circle) fitting on the cross-section using the least squares method instead of the 3D fitting method were evaluated to estimate the radius and center position of the object. Peak P1 in section A–A is higher than peak P2 in section B–B, as shown in Figure7a. The centers of the cross-sections A–A and B–B can be represented by C1 and C2, respec- Photonics 2021, 8, x FOR PEER REVIEW 8 of 14

in Figure 4, a 3D estimation method based on the exact calculation of the diameter was not selected because the width of the diameter distribution is between 1 and 4 μm and silica objects are not necessarily ideal spheres. The measurement results by two-dimensional curve (circle) fitting on the cross-section using the least squares method instead of the 3D fitting method were evaluated to estimate the radius and center position of the Photonics 2021, 8, 112 object. 8 of 13 Peak P1 in section A–A is higher than peak P2 in section B–B, as shown in Figure 7a. The centers of the cross-sections A–A and B–B can be represented by C1 and C2, respectively, as shown in Figure 7b,c. In the curve fitting, the radii of the silica spheres were approximatelytively, as shown 840 in and Figure 880 7nmb,c. in In the the A– curveA and fitting, B–B sections, the radii respectively. of the silica Therefore, spheres were the diametersapproximately of the 840silica and spheres 880 nm centered in the A–A on C1 and and B–B C2 sections, were assumed respectively. to be Therefore,1.68 µm and the 1.76diameters µm, respectively. of the silica Although spheres centeredthe diameter on C1 of and each C2 silica were sphere assumed was to slightly be 1.68 smallerµm and than1.76 theµm, standard respectively. diameter Although shown the in diameter the official of product each silica catalog, sphere the was measurement slightly smaller results than werethe standardwithin the diameter diameter shown distribut in theion official of the productsample. catalog, the measurement results were withinThe the 3D diameterdistance distributionbetween C1 ofand the C2 sample. in the y- and z-directions by curve fitting of the A–A andThe B 3D–B distancecross-sections between was C1 calculated and C2 in using the y the- and 3Dz coordinates-directions by of curvethe measurement fitting of the resultsA–A and based B–B on cross-sections only x-direction was information. calculated using the 3D coordinates of the measurement resultsAs shown based onin onlyFigurex-direction 7a, the distances information. in the x- and y-directions were 170 nm and 1.7 µm, respectively.As shown inAs Figure shown7a, in the Figure distances 7b,c, the in height the x- difference and y-directions between were the two 170 centers nm and µ was1.7 110m, nm. respectively. Therefore, As the shown 3D distance in Figure between7b,c, the heightC1 and difference C2 was approximately between the two 1.71 centers µm. was 110 nm. Therefore, the 3D distance between C1 and C2 was approximately 1.71 µm. In In addition, the center-to-center distance estimated using the sum of the radii of the two addition, the center-to-center distance estimated using the sum of the radii of the two silica silica spheres was 1.72 µm. As the distance between the centers estimated by the results spheres was 1.72 µm. As the distance between the centers estimated by the results was was 1.71 µm, the two silica spheres were considered to be in contact with each other. 1.71 µm, the two silica spheres were considered to be in contact with each other. Therefore, Therefore, it is possible to observe the shape of microstructures in contact with each other, it is possible to observe the shape of microstructures in contact with each other, if the if the diffraction limit was not exceeded. diffraction limit was not exceeded. In this study, silica spheres distributed in different regions, as shown in Figure 5a, In this study, silica spheres distributed in different regions, as shown in Figure5a, were measured in other bright areas, similar to Area-A shown in Figure 5a. were measured in other bright areas, similar to Area-A shown in Figure5a. As described above, this method can be used to observe the silica sphere existing at the As described above, this method can be used to observe the silica sphere existing at imagingthe imaging position position of the of lens the at lens a field at a depth field depth of ±692 of nm.±692 The nm. measurement The measurement results results for each for brighteach brightspeckle speckle pattern pattern area are area show aren shown in Figure in Figure 8. 8.

4.5 1.5 r = 0.60 m] r = 1.46

b1 b1  c1 c1

y [ y r = 0.82 0 m] (c) x [m] 3.0 2  2 b b (c) [ y (b) (d) 0.5 r = 0.88

b3 b3 m] r = 0.50  (a) d1 d1 0 x [m]2.0 [ y 0 x [m] 2.5 (b) (d) Figure 8. Measured results: (a) Measurement area in the speckle interferogram (same image of Figure 8. Measured results: (a) Measurement area in the speckle interferogram (same image of Figure 5a); (b) Measured shape in area; (c) Measured shape in area; (d) Measured shape in area. Figure5a); ( b) Measured shape in area; (c) Measured shape in area; (d) Measured shape in area. As shown in Figure 8b, three silica spheres were connected. The radius of each sphere As shown in Figure8b, three silica spheres were connected. The radius of each sphere was calculated by curve fitting in the same manner as in Figure 7c. The diameters of the was calculated by curve ﬁtting in the same manner as in Figure7c. The diameters of the threethree spheres spheres were were approximately approximately 1.3 1.3–1.8–1.8 µm.µm. In In addition, addition, the the estimation estimation of of the the radius radius of of thethe three three silica silica spheres ledled toto the the assumption assumption that that the the spheres spheres are are in contact in contact with with each each other, other,similar similar to Figure to Figure7a. 7a. Furthermore, a single silica sphere can be measured, as shown in Figure8c,d. The measurement results of a silica sphere with a large diameter of approximately 3 µm are shown in Figure8c. Figure8d shows the measurement results of a silica sphere with a small diameter of approximately 990 nm. Therefore, this method can be used even when randomly distributed microstructures are in contact with each other. Furthermore, the speckle pattern shown in Figure5a has bright areas in addition to the bright regions shown in Figures7 and8. The diameters of 24 microspheres in these Photonics 2021, 8, x FOR PEER REVIEW 9 of 14

Furthermore, a single silica sphere can be measured, as shown in Figure 8c,d. The measurement results of a silica sphere with a large diameter of approximately 3 µm are shown in Figure 8c. Figure 8d shows the measurement results of a silica sphere with a small diameter of approximately 990 nm. Therefore, this method can be used even when randomly distributed microstructures Photonics 2021, 8, 112 are in contact with each other. 9 of 13 Furthermore, the speckle pattern shown in Figure 5a has bright areas in addition to the bright regions shown in Figures 7 and 8. The diameters of 24 microspheres in these 18 bright regions were measured by this method. As a result, the average value of the diam- eter18 was bright 1.85 regions μm and were the standard measured deviation by this method. was 0.633 As μm. a result, the average value of the µ µ diameterThe t-distribution was 1.85 m [29] and was the used standard to a statistical deviation test was of 0.633 the differencem. between the av- The t-distribution [29] was used to a statistical test of the difference between the erage values of the catalog data (2 μm) and the measured results. The t-distribution value average values of the catalog data (2 µm) and the measured results. The t-distribution can be calculated as (2.0–1.85)/(0.633/240.5) = 1.16, since the degrees of freedom are 23. The value can be calculated as (2.0–1.85)/(0.633/240.5) = 1.16, since the degrees of freedom are level of significance is set as 10%. From the t-distribution table, since it can be referenced 23. The level of signiﬁcance is set as 10%. From the t-distribution table, since it can be that t(23,10%) is 1.714 (>1.16), it can be confirmed that the measured average of this referenced that t(23,10%) is 1.714 (>1.16), it can be conﬁrmed that the measured average of method is clearly the same as the average value in the catalog data. this method is clearly the same as the average value in the catalog data. From the above results, it can be concluded that this method can measure the diam- From the above results, it can be concluded that this method can measure the diameter eter of microspheres, provided that the diffraction limit is not exceeded. of microspheres, provided that the diffraction limit is not exceeded.

3.2.3.2. Shape Shape Measurement Measurement of ofObjects Objects with with Random Random 3D 3D Distribution Distribution beyond beyond the the Diffraction Diffraction Limit Limit WhenWhen the the measured measured object object does does not not exceed exceed the the diffraction diffraction limit, limit, the the proposed proposed method method cancan be be used used to tomeasure measure the the shape shape of ofa sphere, a sphere, even even if ifthe the spheres spheres are are in in contact, contact, as as shown shown inin the the previous previous section. section. WhenWhen the the size size of of the the measured measured object is smallersmaller thanthan the the diffraction diffraction limit, limit, the the possibility possi- bilityof measuring of measuring the shapethe shape of a distributedof a distributed object object in 3D in is 3D investigated is investigated using using a silica a silica sphere spherewith awith standard a standard diameter diameter of 700 of nm 700 (diameter nm (diameter distribution: distribution: 0.2–5 0.2µm).–5 µm). TheThe diameter diameter of ofthe the silica silica sphere sphere was was approximately approximately 700 700 nm, nm, according according to tothe the SEM SEM imageimage in inFigure Figure 4b.4b. For For the the optical optical system system shown shown in in Figure Figure 2,2 ,the the diffraction diffraction limits limits were were 523523 and and 1119 1119 nm nm in inthe the cases cases without without and and w withith the the aperture, aperture, respectively. respectively. Therefore, Therefore, the the shapeshape measurement measurement can can be be performed performed under under two two different different measurement measurement conditions conditions of of the the diffractiondiffraction limit limit by by measuring measuring the the same same sample. sample. The The results results obtained obtained with the samesame samplesam- plecan can be be then then compared. compared. TheThe speckle speckle pattern pattern obtained obtained without without the the aperture aperture is isshown shown in inFigure Figure 9a,9a, in in which which AreaArea-B-B is isdefined deﬁned as as the the bright bright region region of of the the speckle speckle pattern. pattern. The The area area surrounded surrounded by by the the brokenbroken line line in inthe the enlarged enlarged view view was was analyzed analyzed using using this this method. method. Carrier Carrier fringes fringes can can be be observedobserved diagonally diagonally in in an an enlarged enlarged view. view. The The shape shape measurement measurement results areare presentedpresented in inFigure Figure9 c.9c. The The contour contour map map of of Figure Figure9c 9c is is shown shown in in Figure Figure 10 10aa. .

y y Area-B Area-C x x

y 15.0 m 15.0 m y 1 m (a) (b) 1 m 200 200

100 100

0 300 0 300 y [nm] x [nm] x [nm] y [nm] 700 0 700 0 (c) (d)

Figure 9. Measured 3D shape of 700 nm sphere at the same position: (a) Speckle pattern without aperture (diffraction limit: 523 nm); (b) Speckle pattern with aperture(diffraction limit: 1119 nm); (c) Phase map of measured 3D shape without aperture in Area-B; (d) Phase map of measured 3D shape with aperture in Area-C.

The cross-sectional shape (C-C), which is parallel to the x-direction, including vertex P3 in Figure 10a, is shown in Figure 10b. The radius and center position of the silica sphere Photonics 2021, 8, 112 10 of 13

calculatedForests 2021, 12, x for FOR thisPEER REVIEW cross-section by the curve ﬁtting process shown in Figure7 were 390 nm3 of 3 and C3 (10, −300), respectively. Hence, the silica object can be deﬁned as a sphere with a diameter of 780 nm.

Figure 10. Measured 3D shape of 700 nm sphere at the same position: (a) Contour of measured 3D shape without aperture in Area-B; (b) Measured 3D shape without aperture in Area-B; (c) Contour of measured 3D shape with aperture in Area-C; (d) Measured 3D shape with aperture in Area-C.

This calculation is performed by the flowchart shown in Figure6 as well as the results shown in Figure7. These results also show that this method can be used to perform measurements if the size of the measured object does not exceed the diffraction limit. Furthermore, the diffraction limit was changed to 1119 nm by inserting the aperture in front of the objective lens of the optical system. The sphere in Area-C shown in Figure9b was measured using an optical system that can measure the same sample at the same location after changing the diffraction limit conditions. Area-C in Figure9b is at the same location as Area-B in Figure9a. However, the speckle pattern in Figure9b is generally darker and more ambiguous than that shown in Figure9a. The 3D shape obtained by analyzing Area-C is shown in Figure9d. Figure 10c shows a contour map of the results shown in Figure9d. The results shown in Figure9c,d are compared. In the same manner as that for the results (C-C) in Figure 10b, the shape of the cross- section (D–D) centered on P4 in Figure 10c was estimated by curve fitting to determine the radius. A radius of 390 nm and center position C4 (20, −290) were estimated by curve fitting, as shown in Figure 10d. The results are approximately in agreement with the results shown in Figure 10b. This confirms the effectiveness of the proposed method, which can be used to measure the shape of a fine structure for a randomly distributed even if the diffraction limit is exceeded. In addition, the diameters of the microspheres at six areas in the speckle pattern shown in Figure9b were measured, although the insertion of the aperture reduced the number of bright regions in the speckle pattern and made data collection difficult. The average value of diameters was 1.09 µm, and the standard deviation was 776 nm. The statistical test using t -distribution [29] was also performed to compare the measured average value with the catalog data (700 nm). In calculation, t = (1091–700)/(776/60.5) = 1.236 was obtained. The level of significance is set as 10%. From t-distribution table, it can be checked as t(5,10%) is 2.015. Since t(5,10%) = 2.015 > 1.236, it can be confirmed that the average of the measured diameters is equivalent to the average value in the catalog data. From the above results, it can be also concluded that this method can measure the diameter of microspheres, provided beyond the diffraction limit. Forests 2021, 12, x FOR PEER REVIEW 3 of 3

Photonics 2021, 8, 112 11 of 13

4. Discussion The above discussion conﬁrms the possibility of measuring the shape of microstructures beyond the diffraction limit of the objective lens by using scattered light as illumination light. Next, the logical mechanism of the measurement process is analyzed. In an optical system based on the concept of perfect imaging [21], scattered lights are emitted from a point Pc on the object surface when illumination light is applied to a rough object surface, as shown in Figure 11.

Figure 11. Model of measurement process.

Some of the reflected scattered lights converge to Pc’ on the image plane of the lens after passing through random positions in the lens aperture. Therefore, when the measured object is set farther than the focal length of the lens on the optical axis, the reflected light from the point (Pc) on the surface of the measured object passes through the lens and converges at the corresponding point (Pc’) of the surface of the measured object. The light emitted from one point on the object surface is focused on the conjugate point considering a perfect optical system. In this case, if the higher-order diffracted light from the measurement object passes through the lens and reaches the image plane, a clear focused image is observed, in accordance with Abbe’s imaging theory [21]. However, when a fine structure beyond the diffraction limit of the objective lens is observed, higher-order diffracted light extends to the outside of the aperture of the lens and cannot pass through the lens, as shown in Figure 11. Therefore, such light cannot reach the image plane. Thus, a focused image could not be captured. According to traditional image theory [21], this imaging process can be attributed to an analysis method based on the formation of a light intensity distribution called an image. When the measurement object is illuminated using scattered light, it is considered that a part of the scattered light from the measured object can pass through the lens aperture, even if the measurement object has a fine structure. As shown by the broken line after scattering at Pc in Figure 11, coherent light passing through various paths interfere with each other. Such scattered light information from each point on the measurement surface is aligned on the conjugate point of the measured surface under the concept of perfect optics. In other words, because the speckle pattern reaches the image sensor through the lens under the perfect optical system, it is assumed that two-dimensional phase information (i.e., shape information) that has a one-to-one correspondence with the position of the measured surface is spatially aligned in a regular manner on the image sensor. Because this method uses phase information, it does not require higher-order diffracted lights for shape measurement analysis, as in the conventional imaging theory based on light Photonics 2021, 8, 112 12 of 13

intensity distribution. If a light with phase information reaches each pixel of the camera through the lens, the phase of the light on each pixel is analyzed with a high resolution, and two-dimensional shape information can be detected on the image sensor using speckle interferometry. If the measured object is laterally shifted by δx and the phase of the speckle from the neighbor point (Pc + δx) of the measured surface can be accurately detected, the two-phase information (φPc before the lateral shift and φPc + δx after the lateral shift) at Pc’ can be sequentially detected according to the principle of the 3D shape measurement. The difference (∆φ) of these phases is equal to the phase value at Pc’ of the specklegram. This specklegram can be calculated between the speckle patterns captured before and after the lateral shift. The derivative value of the shape at Pc can be obtained by divid- ing the difference in phase by the lateral shift δx. By this operation, the two-dimensional distribution of the derivative of the phase regarding the shape of the measured object can be constructed by the computer. This information has a spatial one-to-one correspondence with the surface of the measured object. The original phase distribution f(x) can be reconstructed by integrating the derivative value in the x-direction. As stated above, it is considered that the 3D shape of the ﬁne structure exceeding the diffraction limit can be analyzed by this method without the inﬂuence of diffraction and without using the concept of image formation, which is based primarily on intensity distribution. In addition, the scattered light with multiple ray vectors in various directions is indispensable as illumination light to perform this operation.

5. Conclusions Although it has been reported that the shapes of periodic microstructures can be measured using this method, its usefulness for measuring randomly distributed 3D micro- objects has not been demonstrated. In this paper, the feasibility of measuring randomly distributed 3D objects that exceed the diffraction limit with only one speckle pattern is discussed. Verification experiments were performed using silica microspheres with standardized diameters. The possibility of the shape measurement of the fine objects randomly distributed in 3D was discussed using only a single speckle pattern, even if the size of the object exceeds the diffraction limit. As results of the test based on the statistical processing between catalog data and measured results, it is confirmed that the method can be applied to measure 3D structures randomly distributed in space. In addition, the hypothesis of the measurement physical mechanism that the combi- nation of the concept of perfect optics and the high-resolution phase detection technique using speckle interferometry with scattered light as illumination light enables the 3D shape measurement of microstructures beyond the diffraction limit was discussed. To verify this hypothesis, it is thought that further in-depth investigation and experimental results in wide science fields will be required. In a future study, the limit of the measurement resolution of this method using scattered light will be investigated. The limit of the measurement resolution of this method depends on many parameters, including the lateral shift, which has already been discussed [15]; characteristics of the optical system, particularly the resolution of the lens; imaging resolution depending on the number of pixels and pixel size of the camera; speckle size; optical characteristics of the scattered light; and number of bits during the analysis by the computer. Furthermore, the capabilities of the arithmetic system and the influence of the measurement environment must be considered. These issues will be investigated in detail using experiments and computer simulations.

Funding: This work was supported by JSPS KAKENHI grant Number 20H02165. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Photonics 2021, 8, 112 13 of 13

Acknowledgments: Y. Arai thanks S. Yokozeki in Jyouko Applied Optics Laboratory for help discussion of this work. Conﬂicts of Interest: The author declares no conﬂict of interest.

References 1. Sirohi, R.S. Speckle Metrology; Marcel Dekker: New York, NY, USA, 1993; pp. 99–234. 2. Cloud, G. Optical Methods of Engineering Analysis; Cambridge University Press: New York, NY, USA, 1995; pp. 395–476. 3. Malacara, D. Optical Shop Testing; John Wiley & Sons: New York, NY, USA, 1992; pp. 501–652. 4. George, N.; Jain, A. Speckle reduction using multiple tones of illumination. Appl. Opt. 1973, 12, 1202–1212. [CrossRef] 5. Arai, Y.; Yokozeki, S. Improvement of measuring accuracy of spatial fringe analysis method using a Kalman filter and its application. Opt. Eng. 2001, 40, 2605–2611. 6. Arai, Y. Electronic Speckle Pattern Interferometry based on spatial information using only two sheets of speckle patterns. J. Mod. Opt. 2014, 61, 297–306. [CrossRef] 7. Arai, Y. Pre-treatment for preventing degradation of measurement accuracy from speckle noise in speckle interferometry. Measurement 2019, 136, 36–41. [CrossRef] 8. Leendertz, J.A. Interferometric displacement measurement on scattering surfaces utilizing speckle effect. J. Phys. E 1970, 3, 214–218. [CrossRef] 9. Butters, J.N.; Leendertz, J.A. Speckle Pattern and Holographic Techniques in Engineering Metrology. Opt. Laser Technol. 1971, 3, 26–30. [CrossRef] 10. Macovski, A.; Ramsey, S.D.; Schaefer, L.F. Time-Lapse Interferometry and Contouring using Television System. Appl. Opt. 1971, 10, 2722–2727. [CrossRef][PubMed] 11. Nakadate, S.; Yatagai, T.; Saito, H. Electronic speckle pattern interferometry using digital image processing techniques. Appl. Opt. 1980, 19, 1879–1883. [CrossRef][PubMed] 12. Burke, J.; Helmers, H.; Kunze, C.; Wilkens, V. Speckle intensity and phase gradients: Influence on fringe quality in spatial phase shifting ESPI-Systems. Opt. Commun. 1998, 152, 144–152. [CrossRef] 13. Bhaduri, B.; Mohan, N.K.; Kothiyal, M.P.; Sirohi, R.S. Use of speckle phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS). Opt. Exp. 2006, 14, 11598–11607. [CrossRef] 14. Arai, Y. Three-dimensional shape measurement beyond the diffraction limit of lens. Modern Opt. 2018, 65, 1866–1874. [CrossRef] 15. Arai, Y. Three-Dimensional shape measurement beyond diffraction limit for measurement of Dynamic Events. In Progress in Optomechatronic Technologies; Springer Proceedings in Physics; Springer: Berlin/Heidelberg, Germany, 2019; Volume 233. [CrossRef] 16. Arai, Y. Consideration of existence of phase information of object shape in zeroth-order diffraction beam using electromagnetic simulation with aperture in front of object. J. Mod. Opt. 2020, 67, 523–530. [CrossRef] 17. Kumar, N.; Petrik, P.; Ramanandan, G.K.; El Gawhary, O.; Roy, S.; Pereira, S.F.; Coene, W.M.J.; Urbach, H.P. Reconstruction of sub- wavelength features and nano-positioning of gratings using coherent Fourier scatterometry. Opt. Express 2014, 22, 24678–24688. [CrossRef][PubMed] 18. Madsen, M.H.; Hansen, P.E. Imaging scatterometry for flexible measurements of patterned areas. Opt. Express. 2016, 24, 1109–1117. [CrossRef] 19. Kumar, N.; Cisotto, L.; Roy, S.; Ramanandan, G.K.P.; Pereira, S.F.; Urbach, H.P. Determination of the full scattering matrix using coherent Fourier scatterometry. Appl. Opt. 2016, 55, 4408–4413. [CrossRef] 20. Garnaes, J.; Hansen, P.E.; Agersnap, N.; Holm, J.; Borsetto, F.; Kühle, A. Profile of a high-aspect-ratio-grating determined by spectroscopic scatterometry and atomic-force microscopy. Appl. Opt. 2006, 45, 3201–3212. [CrossRef][PubMed] 21. Born, M.; Wolf, E. Principles of Optics, 7th ed.; Cambridge University Press: Cambridge, UK, 1999; pp. 199–201, 461–476. 22. Arai, Y.; Shimamura, R.; Yokozeki, S. Dynamic out-of-plane deformation measurement using virtual speckle patterns. Opt. Lasers Eng. 2009, 47, 563–569. [CrossRef] 23. Arai, Y. Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method. Measurement 2016, 91, 582–589. [CrossRef] 24. Arai, Y. Influence of error sources in speckle interferometry using only two speckle patterns. Opt. Eng. 2016, 55, 124101. [CrossRef] 25. Takeda, M.; Ina, H.; Kobayashi, S. Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. J. Opt. Soc. Am. 1982, 72, 156–160. [CrossRef] 26. Pavillon, N.; Seelamantula, C.S.; Kühn, J.; Unser, M.; Depeursinge, C. Suppression of the zero-order term in off-axis digital holography through nonlinear filtering. Appl. Opt. 2009, 48, H186–H195. [CrossRef][PubMed] 27. Arai, Y.; Yokozeki, S.; Yamada, T. High-speed two-dimensional fringe analysis using frequency demodulation. Opt. Eng. 1997, 36, 2489–2495. 28. Arai, Y.; Hirai, H.; Yokozeki, S. Electronic speckle pattern interferometry based on spatial fringe analysis method using two cameras. J. Mod. Opt. 2008, 55, 281–296. [CrossRef] 29. McCollough, C. Statistical Concepts; Mcgraw-Hill: New York, NY, USA, 1963; pp. 223–244.