(Open Access) Correlation and convolution filtering and image processing for pitch evaluation of 2D micro- and nano-scale gratings and lattices. (2017) | Xiaomei Chen

Q: What contributions have the authors mentioned in the paper "Correlation and convolution filtering and image processing for pitch evaluation of 2d micro- and nano-scale gratings and lattices" ?

The authors have mathematically explicated and experimentally demonstrated how a correlation and convolution filter can dramatically suppress the noise that coexists with the scanned topographic signals of 2D gratings and lattices with 2-dimensional ( 2D ) perspectives. To realize pitch evaluation, the true peaks ’ coordinates have been precisely acquired after detecting the local maxima from the filtered signal, followed by image processing.

Correlation and convolution filtering and image

processing for pitch evaluation of 2D micro- and

nano-scale gratings and lattices

XIAOMEI CHEN

1,*

, LUDGER KOENDERS

, SIMON PARKINSON

Centre for Autonomous & Cyber-Physical System, Cranfield University, Cranfield MK43 0JR, UK

Surface Metrology Department, Physikalisch-Technische Bundesanstalt(PTB), Bundesallee 100, 38116 Braunschweig, Germany

School of Computing, University of Huddersfield, Queensgate, HD1 4DH Huddersfield, West Yorkshire, UK

*Corresponding author: xiaomei.chen@cranfield.ac.uk

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX

We have mathematically explicated and experimentally demonstrated how a correlation and convolution filter can

dramatically suppress the noise that coexists with the scanned topographic signals of 2D gratings and lattices with

2-dimensional (2D) perspectives. To realize pitch evaluation, the true peaks’ coordinates have been precisely

acquired after detecting the local maxima from the filtered signal, followed by image processing. The combination

of 2D filtering, local-maxima detecting and image processing make up the pitch detection (PD) method. It is

elucidated that the pitch average, uniformity, rotation angle and orthogonal angle can be calculated using the PD-

method. This has been applied to the pitch evaluation of several 2D gratings and lattices, and the results are

compared with the results of using the CG- and FT-method. The differences of pitch averages which are produced

using the PD-, CG- and FT-methods are within 1.5 pixels. Moreover, the PD-method has also been applied to detect

OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (100.0110) Imaging processing; (070.6110) Spatial filtering; (180.0180)

Microscopy.

http://dx.doi.org/10.1364/AO.99.099999

NOMENCLATURE

Correlation

and convolution;correlation or convolution

XYZ

XOY

3D or 2Dcoordinate

systemof samplesurface

xyzorxoy 3D/2D coordinates of measuring instrument, e.g. an SPM

The angles thatXOY plane tilts relative tox-and y-axis of

xyzsystem

, P

Pitches ofa 2D grating or lattice, and periodsofits

topographicsignal

(

)

XYZ

system

, P

Projections of P

and P

in xoy plane: P

⋅

cos

⋅cos

∆x,∆y

Scanning step sizesof an SPMin x- and y-axis

, p

Computer-sampleddata numberswithin P

and P

periods

Pq,Pr Periodsof a half2Dsinusoidal waveform

∆

q ,

∆

Element intervalsofa half2Dsinusoidal waveform

, p

Element numberswithin PqandPr periods

1. INTRODUCTION

The pitch described in this paper is the distance between adjacent

similar structural features of one-dimensional (1D) and two-

dimensional (2D) gratings and lattices on surfaces. In nanometer

metrology and measurement, the International Organization for

Standardization (ISO) stipulates 1D and 2D gratings and lattices in

several documents to calibrate diverse microscopes and instruments

after metrologically verifying the pitch-related parameters, such as

pitch average, pitch uniformity, etc. Microscopes and instruments

include a family of scanning probe microscopes (SPM) [1], scanning

electron microscopes (SEM) [2,3], various optical microscopes and

contact stylus instruments that are used for areal surface roughness

measurement [4,5]. Usually, metrological atomic force microscopes

(AFM) and nano-measuring machines (NMM) [6-8] implement the

metrological verification. This typically includes two steps: acquiring

the three-coordinate topographic signal in raster-scan mode, and

afterwards evaluating the pitch-related parameters according to a

pitchevaluationmethod.

Beside the center-of-gravity (CG) method [9,10] and Fourier-

transform-based (FT) method [10], another pitch evaluation method

of 1D gratings based on a 1D correlation filter has been previously

published [11,12]. A half 1D sinusoidal waveform sequence with

period P

is taken as a correlation filter. When it cross-correlates with a

1D grating topographic signal with period P, the noise can be greatly

suppressed if P

≈P. After correlation filtering, the distance between

any two adjacent waveform peaks, along the direction perpendicular

to 1D grating lines, is one pitch. The method was described as the peak

of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited.

This is the Author Accepted Manuscript issued with: Creative Commons Attribution Non-Commercial License (CC:BY:NC 4.0).

The final published version (version of record) is available online at DOI:10.1364/AO.56.002434.

Please refer to any applicable publisher terms of use.

Applied Optics, 2017, Volume 56, Issue 9, pp2434-2443

DOI:10.1364/AO.56.002434

detection (PD) method. The pitch average, uniformity and rotation

anglearound z-axiscan be calculated using the PD-method.

It has always been cumbersome to evaluate the pitches of 2D

gratings and lattices based on CG- and FT-methods. The 2D gratings

and lattices (defined by XOY plane) are mounted on the stage (defined

by xoy plane) of the measuring instrument to be subsequently raster-

scanned into images. When raster-scanning only a cluster of 2D grating

or lattice features into image, we will find that the grating and lattice

structures are always orientating an unknown

angle around z-axis

relative to the xoy plane. The unknown

angle, plus the accompanied

noise, will make the CG- and FT-method performance more

demanding and less correct. In order to avoid the

angle, the strategy

is to make sure that the

angle in the xoy plane is minimized to zero (

≈ 0°). To achieve this, the 2D grating or lattice that has been loaded

onto the xoy plane, needs to be located, orientated, image-scanned and

image-analyzed to determine if the raster-scan lines are parallel to any

assumed line that passes through a series of gravity centers of the 2D

grating or lattice. The above actions have to be iteratively repeated

until

≈0°. Regardless of the size, the raster-scan area is indispensable

[13] for metrologically verifying a 2D grating or lattice on a

metrological AFM or a NMM, as well as applying it as a standard

material to metrologically calibrate an SPM [14], and to map its errors

[15,16] in accordance with ISO standards [17]. Except for CG- and FT-

methods, there is an absence of literature addressing the problem in

2D pitch evaluation methods. Therefore, our intention is to apply the

PD-method to the pitch evaluation of 2D gratings and lattices by

suppressing the noise and making the

angle known, so that the CD-

and FT-method can be precisely fulfilled with decreasing workload.

Commercially available software [18] has taken a unit cell of

topographic signals from 2D gratings and lattices as the template to

calculate the 2D correlation average. In this paper, a half 2D sinusoidal

waveform is proposed as a template for all features of 2D gratings and

lattices. It can achieve the equivalent impact and high credibility for

analyzing the images and topographic signals containing repeated 3D

structural features. Furthermore, the template as a CC filter can

dramatically suppress the noise and greatly improve the signal-to-

noise-ratio (SNR), consequently the positions and orientations of 3D

features can be precisely characterized andmeasured.

For identifying positions and locations of the repeated structural

features on diverse surfaces [19-22], others have applied a grayscale

threshold segmentation to binarize images, and edge and centroid

detection to extract the borders and locate the centers. In contrast, we

will introduce the 2D CC filter and the peak detection based on local-

maxima detecting and image processing for identifying the peak

positions of 2D gratings and lattices. To the best of our knowledge, the

binary and ternary image reconstruction procedure presented in the

paper is unique. Finally, the mathematical explanation of the 2D CC

filtering, as well as the practicalalgorithm to determine the periods of a

2D half sinusoidalwaveform templatehavebeenannexed.

2. 2D SINUSOIDAL GRATING

A. Topographic and coexisted signals

When an SPM or a scanning tunneling microscope (STM) raster-

scans a 2D sinusoidal grating along two orthogonal direction x and y at

the step size ∆x and ∆y, it crosses the X- and Y-pitches, P

and P

, of the

2D sinusoidal grating with an unknown

angle. The raster-scanned

2D signal F(x,y) (in such physical units as length, voltage, current, etc.),

against the positions (x,y) can be decomposed as a 2D sinusoidal

topographic signal f(x,y), a nonlinear drift signal U(x,y) and a noise

signal W(x,y), i.e.

(

)

(

)

(

)

(

)

, , , ,

F x y f x y U x y W x y

= + +

(1)

Coordinate x and y, signal F(x,y), f(x,y), U(x,y) and W(x,y) are all M ×N

matrices in a raster-scan range M∆x × N∆y. An example signal of a 2D

sinusoidal grating with 300 nm nominal pitches, F(x,y), is shown by a

2D intensity graphin Fig. 1(a).

It is supposed that theorigin of XYZ coincideswiththat of xyz. Due to

the existence of a 2D nonlinear drift signal U(x,y), the XOY plane tilts a

angle relative to x-axis and

angle relative to y-axis. If the 2D

sinusoidal topographic signal,inXYZ systemis defined by

2 2

sin sin( )

, .

X Y

P P

f X Y

π π

(2)

It isexpressed inxyz system as

2 ( cos sin ) 2 ( sin cos )

sin sin

cos cos

2 ( cos sin ) 2 ( sin cos )

sin sin

( ),

X x Y y

x y

x y x y

P P

x y x y

P P

f x

π θ θ π θ θ

ϕ ϕ

π θ θ π θ θ

− +

(3)

where, 1) X=(xcos

−ysin

)/cos

and Y=(xsin

+ycos

)/cos

means

that the coordinates first rotate-transform by an

angle from the XOY

coordinates (of the 2D sinusoidalgrating)to the xoy coordinates(of the

measurement instrument), subsequently rotate-transform by

angle

around the x-axisand

angle around the y-axis respectively; 2) The X-

and Y-axes are parallel to the direction of the X-pitch and Y-pitch, P

and P

, respectively; 3) P

cos

and P

cos

mean the

projectionsofP

and P

in the xoy plane.

U(x,y), according to ISO/DIS 11952[17], is presumably caused by

piezo drift or creep in lateral or vertical direction; mechanical stresses

of the sample holders and its fixers; mechanical expansion of the

components such as measurement frame of an SPM. The diminishing

effect on the accurate pitch evaluation can be leveled by rotation-

transforming

and

angles around the x- and y-axis respectively, so

that U(X,Y) ≈0 in XYZ system, which means the drift signal theoretically

does not exist in XYZ system. Mathematically, it is expressed by a 2D

polynomial function in the xyz system whether it has been leveled or

unleveled:

1 1

( ) ( ) ( ) .

K K

x y c a x b y a x b y

R = + + + + +L

(4)

where, c=a

is the content item, a

and b

(i=1,2, …, K) are the

coefficientsof theith order item of variable x and y , respectively.

W(x,y) is given by the amplitude a

x,y

at any raster-scanposition (x,y):

( , ) .

x y

W x y

(5)

B. 2D CC-filtered signals

A half 2D sinusoidal waveform template T(q, r) with Pq and Pr

periods is described by

2 2

( ) sin sin

, .

q r

T q B

P P

π π

(6)

T(q,r) has a matrix of M

×N

elements against a matrix of M

×N

positions (q,r)with intervals ∆qand ∆r.

The correlation or convolution between F(x,y) and T(q,r) is

expressed by

, , , ,

( ) ( ) ( ) ( ).

TF Tf TU TWR R R R

x y x y x y x y

= + +

(7)

(a) (b) (c)

Fig. 1. (a),(b)and (c)shows 2Dsinusoidal grating topographicsignalF(x, y),correlation-filteredsignalR

(x,y) and peaksdetection image in intensity graphs, respectively.

Fig.2. Based on convolution operation, 3D plots of filtered topographic signalR

(x,y) (without normalization) with periods Px=Py=20 pixels; figure a, b, c, d, e and f are

corresponding to Pq=Pr=P/4,P/2,P,2P, 3Pand 4P, respectively .

(x,y), R

(x,y) and R

(x,y) are named as filtered signal,

filtered topographic signal, filtered nonlinear drift signal and noise

residue signal, respectively.

If correlation operator (+) and convolution operator (−) are

combined into one operator (±), R

(x,y), R

(x,y) and R

(x,y) in xyz

system is developed as equation (8), (9) and (10), respectively. The

mathematical developments are listedin Annex A.

2 (

[ ] [ ]sin( [ ])

cos

2 (

sin( [ ])

cos

cos sin )

( , )

sin cos )

Tf q r X q

Y r

x y

C P D P P

x y

R x y

θ θ

= +

⋅ +

−

(8)

1 1

,( ) ( ) ( ) .TU K

K K

J J Jx y a x b y a x b yR = + + + + +L

(9)

1 1

0 0

( ),( )

( ) ,,

T T

M N

k l

x k q y l r

R x Ay a

− −

= =

± ∆ ± ∆

= ⋅

∑ ∑

(10)

where, C[Pq] and

[Pq] are concerned with Pq whilst D[Pr] and

[Pr]

are related with Pr. They are defined by equation (A.3) ∼ (A.6) in the

Annex A. J is the constant item; J

, J

,⋅⋅⋅, J

are the weighted index of

monomial item, quadratic item, …, and Kth order item of the binary

polynomials, respectively. They are explained by equation (A.8)

∼ (A.10) in Annex A.

Compared with equation (3), equation (8) verifies that R

(x,y)

remains a 2D sinusoidal signal. Its periods, P

and P

, are equal to the

periods of f(x,y), though the amplitude has changed to C[Pq]D[Pr] and

phases have shiftedto

[Pq] and

[Pr].

Compared with equation (4), equation (9) interprets that R

(x,y)

still is a nonlinear drift signal. It will not disturb the peaks detection

even if it is unleveled beforehand, or if it is not totally diminished after

leveled.

In contrast to equation (5), equation (10) is the operation of

weighted moving average (WMA) of the noise signal a

x,y

by using a

matrix of data A

k,l

(k=0, 1, ..., M

-1, l=0, 1, ..., N

-1) as the weights. A

k,l

expressed by equation (A.12) in AnnexA.

Therefore, after correlation or convolution (CC) filtering, the noise

signal a

x,y

, from highly dense irregularities, is minimized to a small and

gently changing noise residue signal R

(x,y). Although it can more or

less modulate the amplitude of the R

(x,y) if it is superimposed to the

latter, it does not influence the periodicity of the latter (see Fig. 3

ref.[12]).

An 80×80 2D sinusoidal simulation signal f(x,y) of 1 arbitrary unit

(a.u.) amplitude and 20 a.u. periods (Px=Py=20), with Gaussian white

noise W(x,y) of 0.3 a.u. standard deviation, was taken as an example to

demonstrate how R

(x,y) varies with period Pq and Pr of a half 2D

sinusoidal waveform template T(q,r). T(q,r) has 1 a.u. amplitude and

×N

elements, where M

=Pq/2, N

=Pr/2. Based on convolution

operation, the 3D plots of R

(x,y) are shown in Fig. 2, where 3D plots

markedby a,b, c, d, e and f correspond to Pq=Pr=P/4, P/2, P, 2P, 3P and

4P,which are 5,10, 20, 40, 60 and 80 a.u., respectively.

From a half 1D sinusoidal waveform as 1D cross-correlation filter

[11,12], when a half 2D sinusoidal waveform of Pq and Pr periods

correlates or convolutes with the raster-scanned signal of a 2D

sinusoidal grating of P

and P

periods withnoise, we candeduce:

(1) itcan greatly filter noise if Pq≈P

and Pr ≈P

;

(2) itcannot completely filter noise if Pq <<P

;

(3) it can filter noise but severely modulate the amplitude of signal

(x,y) to make it impossible to distinguish R

(x,y) from

(x,y) if Pq >>P

andPr>>P

Thus, instead of directly detecting the pitches from the raster-

scanned signal F(x,y), the filtered signal R

(x,y) is validated for the

pitch detection if we chose Pq≈P

and Pr≈P

. For 2D sinusoidal grating

signal F(x,y) shown in Fig. 1(a), the correlation-filtered signal R

(x,y) is

exhibited in 2D intensity graph in Fig. 1(b). The practical algorithm on

how to choose Pq and Pr to implement the CC filtering is attached in

AnnexB.

3. 2D LATTICES

A 2D lattice is a repetitive arrangement of 3D features, such as

pillars, hills, holes, dimples, etc. The 3D convex type features have

parallelogram (rectangle, square, diamond, etc.) or circle bottoms and

the 3D concave type features have parallelogram or circletops. Lattices

are fabricated such that the features are arranged in square,

rectangular, hexagonal and oblique array. The arrangement is in

similarity to 2D solid crystalline lattices. Mathematically, they are

described bydifferent analytic functions inside and zero outside the 3D

features.

A. Topographic signals

For a P

- and P

-pitch lattice with any 3D feature in square,

rectangular, hexagonal, and oblique array, the raster-scanned

topographic signal (with P

- and P

-periods) is defined as f(x,y) inside

the 2D waveforms and zero outside. A primitive unit cell witha bottom

or top area A can be defined in the rectangular range G{-P

/2≤x≤P

/2, -

/2≤y≤P

/2} so that the 3D feature waveforms at the origin lies

entirely within the primitive unit bottom or top, where x and y are two

independent real variables in the whole feature array.The topographic

signal of two exemplar square lattices with 3D central-symmetric

features in parallelogram holes and hills is shown in Fig. 3(a) and (b),

respectively.

The topographic signal of a lattice can be developed as a 2D Fourier

series incomplexexponential form:

( , ) [ exp[ ( )].I JIJ

I J

x yf x y A j x y

ω ω

∞

=−∞

= ⋅ +

∑

(11)

where,

= 2πI/P

and

= 2πJ/P

are the angularfrequencyin x- and

y-axes, respectively; A

is the Fourier transformation coefficient given

( , )exp[ ( )]d d ,

( , 1, 2, ).

xI yJ

A f x y j x y

I J

ω ω

= − +

= ± ± ⋅⋅⋅

∫∫

(12)

(a) (b)

Fig. 3. Topographic signals of square lattices with 3D features in parallelogram

holesand hillsin (a) and (b); the correspondent correlation filtered signals are in

To expand Equation (11) as the real form (Annex C), we find that

2D signal f(x,y) consists of four group of 2D sinusoidal signals with

different phase shifts in each group. Each group includes a constant

item (I, J=0) and infinite numbers of 2D sinusoidal signals including a

fundamental (I, J = 1) period P

and P

, and harmonic period P

/I and

/J (I, J = 2, 3, 4,···). Since the amplitude A

decreases sharply with I

and J increasing [25], the sinusoidal waveform in fundamental period

(I,J=1) dominates equation (11). Concerning the square lattices with

such 3D central symmetric features as shown in Fig. 3 (a) and (b),

equation (11) can be simplified as

, 0

( , ) 4 sin( )sin( )

2 2

I JIJ

I J

x yf x y A x y

π π

ω ω

∞

= − −

∑

(13)

B. 2D CC-filtered signals

If a half 2D sinusoidal waveform with period Pq≈P

and Pr≈P

usedtofiltera series of 2D sinusoidal signals of 2D lattice (with periods

/I and P

/J expressed by equation (13)). The filtered 2D sinusoidal

signal with fundamental periods (I, J=1) has the same period P

and P

The filtered 2D sinusoidal signals with harmonic periods, due to

Pq >>P

/I and Pr >>P

/J (I,J=2,3,4···), have been severely modulated.

Moreover, with periods decreasing (i.e. I and J increasing), their

amplitudes sharply dropped. Therefore, when the filtered 2D

sinusoidal signals with fundamental and harmonic periods are

combined into the filtered signal of 2D lattices, the filtered 2D

sinusoidal signal infundamentalperiodsdominates.

The images of the 2D square holes and 2D hills in square arrays in

Fig. 3 (a) and (b) are raster-scanned by different types of AFM in

256×256 pixels. The actual raster-scan ranges are 90µm×90µm and

50µm×50µm, respectively. After correlation-filtered using a half 2D

sinusoidal waveform with periods, p

and p

of 30 pixels and 40 pixels,

the filtered signals are plotted as 2D intensity graphs in Fig. 3 (c) and

(d), respectively.

4. AUTOMATIC PEAK DETECTION

The peak detection in R

(x,y) can be performed as follows. If a data

(m,n) at (m,n) (m=0,1, ..., M-1, n=0,1, ..., N-1) position in M×N matrix

signal is the true peak, it is the local maximum in both row m and

column n. First, two M×N zero matrices B

and B

are constructed.

Based on the algorithm to find the local maxima in a sequence signals

by applying quadratic/parabolic interpolation of three adjacent

samples [23, 24], the following two steps are taken to detect the local

maxima from row vectors and column vectors, respectively.

Subsequently, the new values are assigned to the corresponding

positionsin B

and B

, respectively:

(1) The local maxima of R

(m,n) are detected row by row. If

(m,n) is detected as a local maximum in row m, B

(m,n) is

converted to 1, otherwise it remains 0.

(2) The local maxima of R

(m,n) are detected column by column.

If R

(m,n) is detected as a local maximum in the column n,

Bc(m,n) isconverted to 1, otherwise it remains0.

As a result, B

and B

are dual-value M×N matrices. Apparently,

(x,y) and R

(x,y) do not influence the local maxima detection,

though they are included in R

(x,y). If a data item R

(m,n) is a true

peak, it should be grey-scale 1 in both images, i.e. B

(m,n)=B

(m,n)=1.

However, if it is only a local maxima, either B

(m,n)=1, B

(m,n)=0 or

(m,n)=0, B

(m,n)=1. If B

and B

are merged into a new image G

using logical ‘AND’ or arithmetical ‘add’ of the corresponding pixels, G

consists of 0 and 1 or 0, 1 and 2 values. The former is called binary

image and the latter is called ternary image. Those pixels with grey-

scale 1 in the binary image G

or grey-scale 2 in the ternary image G

are the true peaks. Thus, a ternary images is displayed in dark

background (grayscale=0), colored pixels (grayscale=1) and bright

pixels (grayscale=2). The local maxima (grey-scale=1), which have

disappeared in the binary image, can produce good visual effect in the

ternary image to associate the peaks with the original and filtered

images.

The peaks detection to the raster-scanned signal of the 2D

sinusoidal grating in Fig. 1(a) is shown bythe ternary image in Fig.1 (c).

The peak detection of the signals in Fig.3 is shown in Fig. 4. Where, (a)

and (b) are the ternary images before 2D correlation filtering. They

appear chaotic and disordered due to noise; (c) and (d) are the ternary

images after 2D correlation filtering. The true peaks in bright pixels

with grayscale 2 can be easily extracted from the ternary images Fig. 4

The peak detection process was applied to the raster-scanned

signals of 2D atomic lattices: (1) silicon (111)-7×7 scanned by the

variable temperature scanning tunneling microscope (VT STM) in

30nm×30 nm range and 800×800 pixel density shown in Fig. 5 (a); (2)

HOPG scanned by the VT STM in 10nm ×10 nm range and 150×150

pixel density shown in Fig. 6 (a). As a result, the correlation-filtered

signals (p

=30 and 10 pixels, respectively) and the ternary images

including true peaks and local maxima are shown in Fig. 5 (b) and (c)

as well as Fig 6 (b) and (c), respectively. It is made possible to use

atoms positions and unit cells to detect the directional drift of the

sample, i.e., the motion of the scanner in an STM.

(a)

(b)

Fig. 4. Ternary image (a), (b), (c) and (d) are the peak detection results

correspondingtothe topographic signal (a) and (b),correlationfilteredsignal (c)

and(d)inFig.3, respectively.

(a) (b) (c)

Correlation and convolution filtering and image processing for pitch evaluation of 2D micro- and nano-scale gratings and lattices.

Figures

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References

Error mapping of high-speed AFM systems

On the characterisation of periodic patterns in tessellated surfaces.

A novel pitch evaluation of one-dimensional gratings based on a cross-correlation filter

Exact and reliable overlay metrology in nanoscale semiconductor devices using an image processing method

Real-time cross-correlation filtering of a one-dimensional grating position-encoded signal

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Frequently Asked Questions (1)

Q1. What contributions have the authors mentioned in the paper "Correlation and convolution filtering and image processing for pitch evaluation of 2d micro- and nano-scale gratings and lattices" ?