Accurate submicron edge detection using the phase change of a nano scale shifting laser spot

Optics & Laser Technology 92 (2017) 109–119 Contents lists available at ScienceDirect Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec Full length article Accurate submicron edge detection using the phase change of a nano-scale shifting laser spot MARK ⁎ Hoang Hong Haia, Liang-Chia Chena,b, , Duc Trung Nguyenb, Shyh-Tsong Linc, Sheng Lih Yehd, Ying Yaob a Department of Mechatronics, School of Mechanical Engineering, Hanoi University of Science & Technology, No Dai Co Viet Road, Hanoi, Vietnam Department of Mechanical Engineering, National Taiwan University, 1, Sec 4, Roosevelt Road, Taipei 10617, Taiwan c Graduate Institute of Electro-Optical Engineering, National Taipei University of Technology, 1, Sec 3, Zhongxiao E Rd., Taipei 10608, Taiwan d Department of Mechanical Engineering, Lunghwa University of Science and Technology,, No.300, Sec.1, Wanshou Rd., Guishan District, Taoyuan City 33306, Taiwan b A R T I C L E I N F O A BS T RAC T Keywords: Focus probe Nanopositioning stage Super-resolution Edge detection Diffraction grating Step height Line width Accurate edge detection with lateral super-resolution has been a critical issue in optical measurement because of the barrier imposed by the optical diffraction limit In this study, a diffraction model that applies scalar diffraction theory of Fresnel–Kirchhoff is developed to simulate phase variance and distribution along edge location Edge position is detected based on the phase variation that occurs on the edge with a surface stepheight jump To detect accurate edge positioning beyond the optical diffraction limit, a nanopositioning stage is used to scan the super steep edge of a single-edge and multi-edges submicron grating with nano-scale, and its phase distribution is captured Model simulation is performed to confirm the phase-shifting phenomenon of the edge A phase-shifting detection algorithm is developed to spatially detect the edge when a finite step scanning with a pitch of several tenth nanometers is used A 180 nm deviation can occur during detection when the step height of the detecting edge varies, or the detecting laser spot covers more than one edge Preliminary experimental results show that for the edge detection of the submicron line width of the grating, the standard deviation of the optical phase difference detection measurement is 38 nm This technique provides a feasible means to achieve optical super-resolution on micro-grating measurement Introduction With the design rules and wafer dimensions in the semiconductor or optical data storage manufacturing industry recently reaching 100 nm and 300 mm, respectively, the demand for determining edge position within an accuracy of 10 nm has been increasing [1] At present, investigations on lateral nano-scale super-resolution show that to accurately determine line width measurement with optical scanning technologies, particularly accurate lateral edge detection is required However, the minimum measurable line width is restricted by the classical resolution limit of optical systems The Rayleigh original criterion is used to define the resolution mathematically when the central maximum of one Airy disc lies over the first minimum of the other, in which two measured points that produce Airy discs can just be resolved individually If we assume that the light source is incoherent and a circulate aperture associated with a microscope objective is employed, the Rayleigh criterion diffraction limit can be expressed as follows [2]: ⁎ dx , y = 0.61λ , NA where λ is the wavelength of the light source, and NA is the numerical aperture of the optical system In an optical interferometric system, the batwing effect is a wellknown phenomenon that is observed around a step discontinuity especially for the case of a step height that is less than the coherence length of the light source, in which the height difference between two adjacent measurement points smaller than λ/4 could not be accurately solved It is usually explained as the interference between reflections of waves normally incident on the top and bottom surfaces When the distorted diffraction image is measured, the edge position for purely topographic line width measurement may fail completely below a certain width because both edge minima will merge into one To solve this problem, various novel techniques have been compared with atomic force microscopy (AFM), which measures direct contact between the tip and the edge in contact mode or near collisions at the edge in non-contact mode An exponential fitting algorithm was Corresponding author at: Department of Mechanical Engineering, National Taiwan University, Sec 4, Roosevelt Rd., Taipei, 10617, Taiwan http://dx.doi.org/10.1016/j.optlastec.2017.01.006 Received 23 July 2016; Accepted 11 January 2017 0030-3992/ © 2017 Elsevier Ltd All rights reserved (1) Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Measurement principle developed for edge detection with an expanded uncertainty of ± 2σ less than 15 nm achievable [3] Another AFM measurement strategy was proposed by PTB, in which the 2D gratings are measured in two narrow rectangular areas for determining all desired measurands [4] Meanwhile, in scanning electron microscopy (SEM) images, edge detection is often performed by thresholding the spatial information of a top-down image To increase measurement accuracy, an edge boundary detection technique based on the wavelet framework is proposed to achieve nano-scale edge detection and characterization by providing a systematic threshold determination step [5] Furthermore, an algorithm based on a self-organizing unsupervised neural network learning is developed to classify pixels on a digitized image and extract the corresponding line parameters [6] The technique was demonstrated on the specific application of edge detection for linewidth measurement in semiconductor lithography In comparison with the SEM imaging, the method can achieve an edge detection with a maximum relative discrepancy of 2.5% Other optical systems have been developed and implemented, thereby leading to the potential establishment of advanced imaging systems with a resolution capability that reaches beyond the diffraction limit of several hundreds to less than 100 nm These methods are based on either intensity [7–10] or phase detection [11–17] Yokozeki et al [18] developed an iterative super-resolution imaging process based on the Richardson–Lucy deconvolution algorithm [19] by using standing wave superposition combined with nanopositioning scanning to detect objects below the diffraction limit However, when this method is used to detect nanoparticles in 2D applications, it fails when existing noise is higher than a spatial frequency of ~9.5 µm−1, which is equivalent to 105 nm; also, frequency components above the crossover frequency cannot be recovered [20] Compared with intensity detection, the phase effect has been detected in the edge region since this idea of phase imaging with phase singularities was first explained by Nye et al [21] The concept of super-resolution phase defects was introduced by Tychinsky [12,22,23] A common path interferometric method provides selective edge detection for line structures because polarization difference is localized at structure edges Zhu and Probst [15,16] proposed using an abrupt nonlinear phase variation in a differential interferometer to detect an edge In their scheme, a heterodyne differential interferometer was modified to produce two polarization mixing beams When one beam scans across an edge, an abrupt phase variation close to 180° occurs If the phase difference between two same-frequency beams is adjusted close to 180°, then a sharp phase variation may occur instead of a phase jump Furthermore, the position of the largest slope in the phase variation is related exactly to the relative position between the scanned spot and the edge This phenomenon can be used to determine the edge position with good certainty Therefore, phase jump can be an ideal index for edge location Masajada et al [24] investigated the diffraction effects of focused Gaussian beams that produced a double optical vortex using a nano-step structure fabricated in a transparent medium, which could be improved by a factor of 15 Hence, measuring the phase that provides additional information regarding the microstructure, is useful for reconstructing an object In this research, a technique used a high-magnification Mirau interferometer objective lens, combined with a nanopositioning stage, is developed for locating the edge of subwavelength structures This research aims to achieve the following objectives: (1) to demonstrate the theory of edge location via the phase variation principle based on theoretical diffraction simulation using the Fresnel–Kirchhoff model, (2) to develop an experimental system to verify the theoretical result with the simulation result The positioning accuracy of the line edge is verified by measuring the grating line width which is defined as a distance between two neighboring line edges of a micro grating 2.1 Theory of diffraction model Fig shows the shape of laser point spot on the grating at different scanning positions When the light is reflected entirely by the substrate or grating top surface, the inspecting light can be simply modeled by light reflection However, when the inspecting light spot is partly engaged with the grating structure, it is partly reflected by either the substrate or grating top surface while the grating edge induces light diffraction, so some of light energy is diffracted away from the inspecting beam The behavior of the inspecting beam being interacting with the grating structure in the scanning process is modeled and investigated by the scalar diffraction analysis as follows In this study, the diffraction light amplitude is calculated at a distance that is significantly longer than one wavelength A scalar diffraction analysis using the Fresnel–Kirchhoff integral to describe Gaussian beam scattering from a phase step surface was conducted by Singher et al [25] The diffraction model is described in Fig When a Gaussian beam incident is focused on a conducting surface, the complex amplitude of the Gaussian beam in scalar approximation is given under the TEM00 mode as follows: E (x , y , z ) = E ⎡ ⎧ ⎡ ⎤ ⎛ z ⎞⎤ jk ⎫ ω(0) ⎬ × exp( − jkz )exp⎢j tan−1⎜ ⎟⎥ , exp⎨− r 2⎢ ⎥ + ⎣ ω (z ) ⎦ ⎝ z ⎠⎦ ω (z ) 2R(z ) ⎭ ⎣ ⎩ (2) where E(x,y,z) is the complex amplitude of the diffracted light at a distance z from the aperture plane to the observation plane, and E0(x0,y0,z=0) is the complex amplitude at the aperture plane The functions ω(z ) and R(z) are referred to as the beam waist and the curvature radius of the phase font, respectively ⎛ z2 ⎞ ω 2(z ) = ω 2(0)⎜1 + ⎟ , z0 ⎠ ⎝ R (z ) = z + z0 = z 02 , z πω 2(0) , λ (3) (4) (5) Fig Schematic diagram of the laser spot scanning at the (1) phase step, (2) single line-width, and (3) groove of the grating sample 110 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Fig The Fresnel-Kirchhoff diffraction model through a circular aperture Fig Gaussian beam reflected by the phase step that considers the position of the beam with (a), (b), (c), and (d) to the left of the beam center, (e) at the beam center, and (f), (g), (h), and (i) to the right of the sample (j) 3D beam profile at a position near and at the edge from the laser focal spot as follows: where ω(0) is the minimum beam waist defined to be the point, where the intensity of the beam decreases to 1/e2 of its maximum To test the diffraction between two propagation planes of the step height, the diffracted wave at a z plane can be expressed by substituting the complex amplitude of the Gaussian beam into the Fresnel– Kirchhoff integral by considering the lateral shifting distance Δy (nm) exp( − jkD ) R(x1, y1)E1(x1, y1 + Δy ) a∼ jλD ⎧ −jk ⎫ × exp⎨ [(x − x1)2 + (y − y1)2 ]⎬dx1dy1, ⎩ 2D ⎭ E (x , y , z = D ) = 111 ∬ (6) Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al where E1(x1,y1) is the incidence wave; a ̃ is the aperture diffraction; and R(x1,y1) is the complex transmission or reflection of the aperture, which is assumed to be equal to D is the distance from the aperture plane to the observation plane, and Δy is the position of the beam shifting along the horizontal axis y1 Given that the item for the integral is variance separable for x1 and y1, ∞ E (x , y , z = D ) = a ∞ ∫−∞ exp(−b(x1)2 + cxx1) dx1 × ∫−∞ exp(−b(y1)2 + cyy1) ⎧ ⎫ ⎡ ⎪ jk ⎤⎥⎪ ⎬dy1, × exp⎨ − q 2⎢ + ⎪ ⎢⎣ ω(z ) 2R(z ) ⎥⎦⎪ ⎩ ⎭ (7) where the coefficients a, b, cx, and cy are calculated as follows: a= b= Fig Procedure of measurement approach ⎧⎡ ⎛ z ⎞⎤⎫ exp⎨j ⎢ −kz + tan−1⎜ z ⎟⎥⎬ ⎝ ⎠⎦⎭ ⎩⎣ ⎪ ⎪ ⎪ ⎪ ω(z ) jk jk + + , ω (z ) 2R ( z ) 2D cx = x jk , D (8) (9) (10) Fig Simulation of single-step height scans at different heights (laser spot size is µm, focused on the top, wavelength is 632.8 nm) (a) Phase jump scanning through the edge, (b) edge detection phase response, and (c) zoomed-out view 112 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Fig Simulation of line width scans for different height ( laser spot size is µm, focus on the top, wavelength 632.8 nm (a) phase jump scanning through the edge, (b) the edge detection phase response and zoom out (c), respectively I (x, y, z ) = E (x, y, z )E *(x, y, z ) Table Simulation result of 1.5 µm line width with different heights: 0.095, 0.2, and 0.5 µm edge criterion 2nd derivative of phase height [μm] w [μm] error [%] 0.095 1.574 4.9 cy = − q ω (z ) 0.2 1.491 0.59 The simulation result for the He–Ne laser with wavelength λ=632.8 nm was implemented via MATLAB software A spot was focused on top of the step height h=190 nm The beam waist was calculated via the diffraction limit spot size of the beam ω0=700 nm (the ideal diameter of the focused probe of a He–Ne laser) The radius curvature R(z) =∞ on the left of the edge (y1 < 0), whereas R(z) =2.65 µm for the second half of the plane (y1 > 0) The calculated diffraction pattern with 100 nm scanning step is shown in Fig The simulation result shows that when the Gaussian beam passes through the edge of a step height, it will create two peaks with a deep gap in between them For the changing positions of the beam under various values of the shifting Δy, the calculated diffraction pattern shows that the reflected energy is pushed to the sides by the destructive interference near the step edge When the spot is on the surface, the total light intensity received is shown in Fig 3(a) This intensity will gradually decay when the spot is entirely out of the surface as shown in Figs 3(b) to (h) and will completely disappear in Fig 3(i) The theoretical slope shows that the ideal edge position corresponds to the minimal slope point when the scanning distance from the center point of the laser spot to the edge is equal to the radius of this focus spot size [26] 0.5 1.435 4.35 ⎛ k k⎞ + j ⎜y − q ⎟ , ⎝ D R⎠ (11) and the aforementioned integral is divided into two cases with y1 ≥ and y1 ≤ Thus, 2D diffraction can be performed by simply performing 1D diffraction integrals, which yields substantial performance enhancement Finally, the complex amplitude is expressed as ∞ E (x, y, z = D ) = a1 ∞ + a2 ∫−∞ exp(−b1(x1)2 + cx1x1)dx1 × ∫−∞ exp(−b1(y1)2 + cy1y1) dy1 ∫−∞ exp(−b2(x1)2 + cx2x1) dx1 × ∫0 ∞ exp(− b 2(y1)2 + cy2y1) dy1 (13) (12) From Eq (12), the beam has a Gaussian transverse intensity profile as follows: 113 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Fig Simulation of multi-edges with a height of 100 nm (laser spot size is µm, focused on the top, wavelength is 632.8 nm (a) Phase jump scanning through the edge, (b) edge detection phase response, and (c) zoomed-out view with a constant phase shift δ (t ) =π/2 as follows: Table Simulation results of grating scanning I0i(x, y, t ) = Ibackground (x, y ) + Iamplitude(x, y )cos [ϕi(x, y )] Edge criterion 2nd derivative of phase Grating element w [μm] Error [%] 1.504 0.2 1.489 0.7 1.517 1.13 1.491 0.6 δ (t ) = 0, (15) I1i(x, y, t ) = Ibackground (x, y ) − Iamplitude(x, y )sin [ϕi(x, y )] δ (t ) = + π /2, (16) I2i(x, y, t ) = Ibackground (x, y ) − Iamplitude(x, y )cos [ϕi(x, y )] δ (t ) = + π , I3i(x, y, t ) = Ibackground (x, y ) + Iamplitude(x, y )sin [ϕi(x, y )] δ (t ) = + 3π /2, I4i(x, y, t ) = Ibackground (x, y ) + Iamplitude(x, y )cos [ϕi(x, y )] δ (t ) = + 2π (17) 2.2 Edge detection (18) When Eq (13) is applied, the transverse intensity profile of the point-focused beam that follows the Gaussian distribution of the interferometer is determined from the intensity of a reference beam and an object beam, and can be described by the following equation: I (x, y, t ) = Ibackground (x, y ) + Iamplitude(x, y )cos [ϕ(x, y ) + δ (t )], (19) To calculate the phase change at different nanopositioning scanning distances, a scanning procedure is shown in Fig The unknown phase of object displacement at the ith position ϕi(x, y ) is calculated as follows: (14) where I(x, y,t) is the interference image distribution in the detector Ibackground(x, y) and Iamplitude(x, y) are the background and amplitude light intensities, respectively ϕ(x, y ) and δ (t ) are the measured object phase and phase shift, respectively A five-step phase-shifting algorithm acquires five interferograms ⎞ ⎛ 2(I1i(x, y ) − I3i(x, y )) ϕi = tan−1⎜ ⎟ I ( x , y ) − I ( x , y ) − I ( x , y ) ⎠ ⎝ 2i 4i 0i (20) In 1D scanning, an edge is defined as the position value of the 114 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Fig Optical layout of the developed Mirau interferometer for edge detection transition from a low/high value to a high/low value of the signal function Various edge detection criteria are discussed in [4] In this study, the edge is defined as a point at the location where the phase singularity or jump occurs across the edge Thus, the edge position is detected at the position where the second derivative of the phase difference is exactly at its zero crossing point The edge position xe is determined when G(x) as defined in Eq (21) equals to zero G (x ) = ∂ 2ϕ(x ) ∂x (21) The edge positions are determined by a left rising edge xleft and a right falling edge xright Then, the width of the measured line width is the absolute value difference between both edge positions xleft and xright of the edge To investigate the scanning of a focus laser beam through single-, double-, and multi-edges, the single edge was first detected Fig shows the location of a single edge based on total phase distribution (Fig 5(a)) when scanning at different step heights, namely, 0.095, 0.2, and 0.5 µm When scanning is performed from the left to the right of the edge, the height difference is small Therefore, the light intensity of the laser beam spot under different total heights is not changed In Fig 5(a), when the center of the laser spot is at a lateral position (3 µm), the spot begins to touch the edge of a step, thereby resulting in a decrease in total light intensity Total light intensity remains stable until the center spot touches the edge at a position of up to+3 µm, which is completely out of the edge Then, the obtained phase of each point was calculated using Eq (20) As shown in Fig 5(a), phase distribution will be approximately constant for a small phase shift but will decrease strongly near the edge The position of phase discontinuity jumps from to 180° with a true edge position at zero point In Fig SEM images of the TGZ02 ultra-sharp edge grating standard with a reference hole used in the experiment 115 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Fig 10 TGZ02 measurement: (a) height and (b) width of the pitch from the 2nd derivative of phase measurement Fig 11 TGZ02 measurement: (a) height and (b) width of the pitch from the 2nd derivative of phase measurement provided in Fig 5(c) The results of its phase quadratic differential curve through the zero point will be extremely close to the edge position, thereby representing phase points that are nearly at the edge The shifting to the left position is −0.002715, 0.0031, and 0.00015 µm addition, different step heights will affect phase distribution changes The greater the phase change, the higher the height To clearly distinguish the inflection point of the zero relative position, the second phase derivative is shown in Fig 5(b), with the zoomed-out view 116 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al Fig 12 SEM images of the RS-N grating standard with a pitch of 0.8 µm and a reference hole used in the experiment Fig 14 RS-N grating measurement at 0.8 µm pitch: (a) height and (b) width of the pitch from the 2nd derivative of phase measurement Subsequently, the computation of two edges with a three-step height was scanned The spot radius µm scans from the left to the right of line width 1.5 µm, with one edge of each 0.75 µm from the zero point Fig simulates the scanning edge location of different step heights: 0.095, 0.2, and 0.5 µm Fig 6(a), as a light phase distribution of the two edges, can be observed when the focused beam touches the first edge, which is at the position of −3.75 µm Total light intensity begins to change when the light spot is attenuated in the position ranging from −0.75 to 0.75 µm Then, the spot center covers the two edges Light intensity is reflected as a result of Gaussian spot distribution Given the diffraction at the edge, total light intensity is still not as strong as the initial power, until finally, the spot center scans until the second edge After that, total light intensity starts to increase slowly The edge location was calculated for phase distribution as shown in Fig 6(b), with the zoomed-in view provided in Fig 6(c) The edge is located within the vicinity of 0.7 µm, with lateral positions of −0.787, 0.7456, 0.7174 µm, thereby suggesting that the location offset reaches up to 40 nm of the line width and produces a maximum 4.9% error for a single width (Table 1) The varying height of the line width is a strong Fig 13 RS-N grating measurement at 0.8 µm pitch: (a) height and (b) width of the pitch from the 2nd derivative of phase measurement 117 Optics & Laser Technology 92 (2017) 109–119 H.H Hai et al In the second experiment, the phase scanning of a silicon grating resolution standard RS-N (Simetrics GmbH) with a pitch of 0.8 µm is shown in Fig 12 The profile of the measured AFM data is shown in Fig 13(a), and its calculated second derivative values are shown in Fig 13(b) The experiment result of the submicron sampling scanning phase is shown in Fig 14(b), which is reasonable compared with the AFM measurement when mean deviation from AFM measurement is 17 nm The conventional diffraction limit of this system can be overcome according to Eq (1) In actual experiments, however, the characteristic curve of the intensity response is not too smooth With the inevitable occurrence of noise (e.g., dark noise, which is independent from signal intensity, and shot noise, which is based on the stochastic nature of particle counting) disturbance from different components of the experiment system (e.g., detector, electronic parts, vibration, sampling time, sample properties), obtaining an ideal slope curve to find an accurate zero crossing point is impossible Senoner [27] demonstrated that a reasonable resolution criterion for dip-to-noise ratio should be as follows to separate noiseinduced intensity variations: Table Measured pitch via AFM and the proposed method, where 2w is the grating period Method AFM 2w[μm] a 2.960 Mean deviation from AFM measurement [nm] σ [nm] 2nd phase derivative b 2.912 c 2.912 a 2.9212 b 2.9072 c 2.9078 38.8 48 13.4 38 34 26 impact factor; that is, the higher the step height, the stronger the phase changes near the edge Finally, the simulation result of scanning multiple line widths of a grating that is 100 nm in height and 3.0 µm in width is shown in Fig Phase change across multiple edges of the grating is shown in Fig 7(a) The second derivative of the phase with zero crossing points is shown in Fig 7(b) Then, the difference in line widths is presented in Table Grating pitches are calculated as 2.993, 3.015, and 3.008 µm The average error relative to the ideal line width is 0.65%, which is approximately 10 nm This simulation result can be used to ensure the correctness of the edge detection model D / σNR ≥ 4, (25) where D is the dip between two strips of square wave grating and σNR is the quantified noise based on its standard deviation Therefore, to reduce the attained uncertainty of the edge position, the measurement was repeated 30 times From the repeatability test, the standard deviation of the measured line width was 38 nm using the phase gradient of the edges The result, compared with the reference AFM tapping mode measurement method (Table 3), verified that the different line widths of the grating could be determined accurately Optical system setup To verify the theory presented in this paper, a Mirau phase-shifting interferometry technique with nanoshifting laser focused-beam detection is illustrated in Fig A scanning laser focus probe is used to detect the edge laterally First, a laser light source (He–Ne, wavelength: 632.8 nm) passes through a beam expander and a non-polarizing beam splitter cube (NPBS1) and then propagates into another beam splitter cube (NPBS2) The transmitted beam goes through the Mirau interference microscope objective, which then focuses on the sample The reflected interference beam goes through the tube lens and is recorded by a CCD camera The sample is mounted on an x–y nano-scale piezoelectric transducer stage (100 àmì100 àm) for lateral scanning through steep edges to detect grating edges Meanwhile, another PZT combined with the objective, which allows accurate translation steps in the vertical direction (1 nm resolution), is used to adjust the light focusing on the reference mirror to perform Phase Shifting Interferometry PSI An analyzer is also used in the optical path between the tube lens and the CCD camera to modulate polarization in light intensity control Conclusions In this study, a new method for submicron-resolution sharp edge detection, with line widths and step heights smaller than half-wavelength measurements, is proposed using the total reflected intensity of nearly common path interference focus spot detection assisted with nano-scale scanning The computation of the diffraction model is based on the Fresnel–Kirchhoff integral, which indicates that regardless of whether the edge is detected via total intensity or phase information, an evident periodical variance will be observed A 632.8 nm He–Ne laser with a light spot (radius: µm) was used to scan a double-edged step height of 0.10 µm The simulation result shows that when the line width between the two edges is less than 0.6 µm (20% of the light spot diameter), we cannot resolve the positions of the two edges This research determines the relationship between laser spot size and minimum edge resolution The experiment shows that the edge point and line width with optical super-resolution can be accurately identified via phase gradient, with reasonable accuracy and repeatability within 38 nm We achieve good qualitative agreement between our numerical simulations and the experimental findings Compared with traditional interference microscopes, the developed method can achieve lateral super-resolution for edge detection and is possible for in situ measurement Future research should calculate surface reflection and signal-to-noise ratio, and then predict the efficient physical model to eliminate systematic deviations based on the calculation results Experimental results The results of the line width measurements at an ultra-sharp edge silicon grating TGZ02 (Micromash GmbH) with a mean step height is 104.9 ± 1.2 nm and the pitch is 3.0 µm, which is carried by a nanopositioning state The scanning position starts from the reference hole, as shown in Fig This hole is approximately µm in diameter, 100 nm in depth, and is situated several microns away from the first edge This hole was produced using a Nova-600i focused ion beam instrument (Department of Electrical Engineering, National Taiwan University) The measurement height is 106.2 nm via Bruker Dimension Icon AFM system (Center for Measurement Standards in Industrial Technology Research Institute, Taiwan) as shown in Fig 10(a) The uncertainty analysis of the AFM is based on ISO/IEC Guide 98–3:2008 Its expanded uncertainty for a 95% confidence level is 0.13 nm Meanwhile, line width is measured using a second derivative of the height profile The result of the different line widths of the grating is shown in Fig 10(b) and compared with the proposed measurement shown in Fig 11(b) The experimental and theoretical profiles for phase distribution scanning (Fig 7) are 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waist was calculated via the diffraction limit spot size of the beam ω0=700 nm (the ideal diameter... reduce the attained uncertainty of the edge position, the measurement was repeated 30 times From the repeatability test, the standard deviation of the measured line width was 38 nm using the phase