In this paper, we consider a method for estimating a stripetype defect and the reconstruction of a defectfree L/S type mask used in lithography. Comparing diffraction patterns of defected and defectfree masks, we derive equations for the estimation of the location and size of the defect. We construct an analytical model for this problem and derive closed form equations to determine the location and size using phase retrieval problem solving techniques. Consequently, we develop an algorithm that determines a defectfree mask pattern. An example shows the validity of the equations.
I. INTRODUCTION
An important problem in extreme ultraviolet lithography (EUVL) is the detection and characterization of defects on multilayer masks
[1]
. The fine structures of a mask present features much smaller than the wavelength, hence conventional imaging methods cannot provide images with sufficient resolution.
One proposed approach relies on coherent diffraction
[2
,
3]
. A coherent beam is projected onto the mask, and the diffraction pattern is measured. As a result, the information obtained is not directly the shape of the mask but instead the modulus of its Fourier transform. Phase information is lost, but it is possible to recover it by a phase retrieval method if the sampling frequency is high enough. The principle of the phase retrieval algorithm is to go back and forth between real and frequency domains and apply constraints such as positivity, finite support, or the given Fourier transform magnitude in these two respective spaces. Several phase retrieval algorithms with different types of constraints have been proposed
[4]
. The most widely used is the hybrid inputoutput (HIO) algorithm, as it enables converging to a global minimum. Unfortunately, in the presence of noise, convergence may be very slow, and in practice, solutions tend to oscillate as a function of the number of iterations because separate constraints in the real and Fourier spaces cannot be satisfied at the same time. Several solutions have been demonstrated to make the phase retrieval algorithm more robust to noise, such as the oversampling smoothness method
[5]
, but they still suffer from some limitations. Kim
et al.
[6]
and Kinoshita
et al.
[7]
both successfully applied iterative approaches to the case of defect detection in EUVL.
Another suggested approach is the deterministic method. In this case, control data are needed, such as from a defectfree mask. For example, Taylor proposed an offaxis holography technique by adding a point signal far off the region of the support of the desired signal
[8]
. Fienup has considered the reconstruction of signals having latent reference points
[9]
. Fiddy
et al.
proposed a method using Eisenstein’s criteria to make a twodimensional signal irreducible
[10]
. Podorov
et al.
and GuizarSicario have proposed some methods using a new type of additive reference signals
[11
,
12]
. In
[13]
, deterministic algorithms are developed for the phase retrieval problem where the two sheets of Fourier transform magnitude (FTM) or Fourier intensity are assumed known, i.e., the FTM of a desired unknown signal and the signal obtained by the addition of the unknown desired signal and a reference signal. The reference signal is also known. In
[14]
, the authors have applied a deterministic approach to find a point defect in line and space (L/S) masks.
In this paper, we consider the application of a deterministic algorithm for this problem. First, we assume that the mask has an L/S pattern and that the diffraction pattern of the original defectfree mask is available. Secondly, we assume that the shape of the defect is a single stripe and is located on the left of the mask. This assumption is not critical because the diffraction pattern is preserved against some transforms including translation, especially reflection, in particular timereversal for a onedimensional signal. The goal is to estimate the location and shape of a stripe defect and distinguish it from the diffraction pattern of a defectfree mask. Contrary to iterative approaches, our method enables the determination of the exact shape of a defect, including its position and depth. The reconstruction of the defectfree mask may be important in the sense that we need to check whether the structure of the real mask, which is referred to as a reconstructed defectmask in this paper, has the same structure as the one that we had designed and known structure. Also, although in many cases we have some knowledge on the shape of the defectfree mask, when we use a mask pattern for the first time, we need to determine the exact shape of it to compare whether the real one has exactly the same structure that we had designed and intended to have.
This paper is organized as follows. In Section II, we describe the problem and a mathematical model as a solution. In Section III, we derive equations for determining the values of interest. In Section IV, we show an example demonstrating the validity of the algorithm.
II. PROBLEM DESCRIPTION
We consider a simple stripetype defect that occurred in an L/Stype pattern mask
[1]
.
Figure 1
is a shape and model of such a mask and a defect.
Figure 1(a)
is the shape of a defectfree mask and
(b)
is the crosssection of this mask on the xy plane. We let the mathematical notation of the defectfree mask model be
x
(
n
). For simplicity, we assume that
x
(
n
) consists of two values
x_{mask}
and
x_{sub}
representing the signal values of the line parts and the blank or substrate parts of the mask, respectively. We also assume that the first and the last points, that is, the two boundary values, of the mask pattern have values of
x_{mask}
. Then the defected mask pattern can be represented as
Schematic and mathematical models of the problem. (a) Shape of the mask that we are considering, (b) Cross section of the defectfree mask pattern, (c) Cross section of a pattern with a defect, and (d) Shape of the defect only and its mathematical model.
where
h
(
n
) is a discrete representation of the defect signal and assumed to be represented as a linear combination of impulse signals, i.e.,
where
n_{i}
’s are locations of the defect points and
A_{i}
’s, which are real and positive, are the amount of defect at these points.
Figure 1(c)
is the crosssectional shape of the defected mask and the defect and
(d)
is the shape of the defect and its mathematical model.
For convenience, we assume that all
n_{i}
’s are consecutive, where
n
_{1}
is the first and the smallest and
n_{M}
is the last and the largest. To simplify the derivation of equations, we assume that the defectfree mask has a region of support [0,
N
1] and the defect is located on the left side of the maskall
n_{i}
’s are less than (
N
1)/2. For the case in which
n_{i}
> (
N
1)/2, the same equation and approach can distinguish the defected and defectfree masks. However, in this case, the estimated defectfree mask pattern will be a flipped version of the results for
n_{i}
< (
N
1)/2 with respect to the center of the region of support. We further assume that we have the information about the diffraction pattern of the defectfree pattern
x
(
n
). To derive these values we assume that the Region of Support (ROS) of
x
(
n
) is
R
(
N
) = [0,
N
1] and the ROS of its diffraction pattern is at least twice of that of
x
(
n
). Also, we assume that the diffraction pattern of these mask patterns can be measured in at least twice the ROS of the mask patterns, i.e.,
R
(
N
,
N
) = [(
N
1), (
N
1)]. Finally, we assume that the width of a defect is less than the minimum width of line parts, hence,
n_{M}

n
_{1}
+1<
min
{
W_{mask}
}. The remaining problem is to get the values of
n_{i}
’s and
A_{i}
’s from the given information. In the next section, we consider the derivation of the necessary equations.
III. RECONSTRUCTION
In this section, we derive equations for the calculation of the values of
A_{i}
’s and
n_{i}
’s.
 3.1. Mathematical Preliminaries
3.1. Mathematical Preliminaries We assume that we have a prior knowledge of the diffraction pattern of the defectfree mask pattern. Then, if we take the inverse Fourier transform of the diffraction pattern, we can get the autocorrelation function of the mask pattern signal. In mathematical terms,
Likewise, if we get the inverse Fourier transform of the square of the diffraction pattern of the defected signal
y
(
n
), then we get
Using relationship (1), the autocorrelation function
r_{y}
(
n
) is related to
r_{x}
(
n
), A, and
x
(
n
) as follows:
where
Here,
r_{x}
(
n
) is assumed known and
r_{y}
(
n
) is a measurable quantity. For convenience, we define
r_{xy}
(
n
) as
r_{xy}
(
n
) =
r_{x}
(
n
) 
r_{y}
(
n
). Then from Eq. (5), we get
If we plot this
r_{xy}
(
n
) signal, we get
Fig. 2(b)
if we assume that
x
(
n
) is of the shape
Fig 2(a)
. As we can see that Eq. (7) is composed of 3 terms, the figure is also composed of 3 different figures. The last term
r_{h}
(
n
) is composed of several delta functions around the origin. Each of the first and the second terms of Eq. (7) is composed of the sum of scaled and translated versions of the defectfree mask pattern or its timereversed versions, and these two terms are overlapped around the origin. Based on this, we can decompose the support of the defectfree mask into 3 regions (
Fig. 2(c)
). Region #1 is the region where the 3 terms in Eq. (7) do not overlap with each other. The remaining part is bisected into Region #2 and Region #3.
Decomposition of r_{xy}(n). (a) Schematic model of a defectfree mask. (b) Decomposition of r_{xy}(n), (c) Decomposition of the regions of the mask.
 3.2. Determination of Boundary Valuesxmask
The boundary value
x_{mask}
can be determined from the autocorrelation function of the defectfree signal
x
(
n
). Since we assumed that the two boundary values
x
(0) and
x
(
N
1) have the same value
x_{mask}
, from the autocorrelation function
r_{xy}
(
n
), we get
From this equation, we determined two boundary values
x
(0) and
x
(
N
1) as
x
(0) =
x
(
N
1) =
x_{mask}
.
 3.3. Estimation of the Defect
Since we do not know the exact shape of the defectfree mask signal, we need to devise some ways to determine the location of the defects. From
Fig. 2(b)
, which shows the structure of
r_{xy}
(
n
),
r_{xy}
(
n
) is composed of the superposition of several terms. In addition to the terms
and
there is a term
r_{h}
(
n
), which is composed of several delta functions. Among all of the nonzero points of
r_{xy}
(
n
), only two boundary points are not composed of superposition of several terms. The main idea here is to determine the location and amplitude of defect points and then to cancel their contribution from
r_{xy}
(
n
) to determine the next point. Since the support of
r_{xy}
(
n
) is given as [(
N
1
n
_{1}
), (
N
1
n
_{1}
)] and is symmetric, it is sufficient for us to consider only the right half of the signal. From this, we can determine the location of the first point of the defect, i.e.,
n
_{1}
. Also, since the right most term is given as
A
_{1}
x
(
N
1) and since we know the value
x
(
N
1), we can determine
A
_{1}
, i.e.,
A
_{1}
=
r_{xy}
(
N
1
n
_{1}
) /
x
(
N
1).
The remaining values can be determined as follows. If we subtract
A
_{1}
x
(
N
1)
rect
(
n
_{1}
,
N
1
n
_{1}
) and
A
_{1}
x
(
N
1)
rect
(
n
_{1}

N
+1,
n
_{1}
) from
r_{xy}
(
n
), where
then we can erase the effect of the first term
A
_{1}
x
(
n
+
n
_{1}
), although we do not know the shape of
x
(
n
) exactly as illustrated in
Fig. 3
. As can be seen there, now we can determine the location and amplitude of the next point in the defect. This is valid if the width of the defect is narrower than the minimum width of the absorber of the mask, which we previously assumed. In general, the
i
_th defect value
A_{i}
can be determined from the equation
Figure of r_{xy_temp,1}(n) = r_{xy}(n)  A_{1}x(N1)rect(n_{1}, N1n_{1})A_{1}x(N1)rect(n_{1}N+1, n_{1}).
where
and
index
_{max}
is the maximum index of nonzero values of
r_{xy_temp,i}
(
n
). By calculating these recursively, we can determine all of the points in the defect until the signal
has no values greater than 0.
Consequently, we are able to determine the location and amplitude of all defects. We can represent the defect profile the same as the equation given in Eq. (2).
Figure 4
is the block diagram of the algorithm for estimating the defect signal.
Block diagram of the algorithm for the estimation of the defect.
 3.4. Reconstruction of the Mask Structure
Since in 3.3, we have determined the defect signal, now we can reconstruct the defectfree mask signal
x
(
n
), and therefore the defected signal
y
(
n
) as well, from the data obtained so far. We first start with the defect signal. Since we have estimated the defect signal, we can also determine the autocorrelation function of
h
(
n
),
r_{h}
(
n
), exactly. Now we define another signal
r_{xyh}
(
n
) as
In
Fig. 2(c)
, we have divided the support of the mask pattern into 3 regions. The reconstruction may be done in 3 steps.
 3.4.1. Region #1
The range of Region #1 covers the indices of [
n
_{1}
+
n_{M}
+ 1,
N
1]. Among these, we have already found the boundary value
x
(
N
1). If we look at the support [
n_{M}
+1,
N
1
n
_{1}
] of
r_{xyh}
(
n
), we can see that the right side of Eq. (12) is composed only of the right summation term of
.
Using this equation recursively, we determine the values of x(n) as
for
n
=
N
−2,
N
−3,...,
n
_{1}
+
n_{M}
+1.
 3.4.2. Region #2
The range of Region #2 covers the indices of
n
= 1,2, …,
, where
is the largest integer that does not exceed (
n
_{1}
+
n_{M}
)/2. The values of this region can be determined from the definition of the autocorrelation function of the mask signal x(n):
for
n
= 1,2,…,
.
 3.4.3. Region #3
The values of the mask pattern in Region #3 starting from
+1. to (
n
_{1}
+
n_{M}
) can be determined from the following equation:
for
n
= (
n
_{1}
+
n_{M}
), (
n
_{1}
+
n_{M}
1),…,
+1.
Summarizing these 3 equations, we can reconstruct the entire values of the mask pattern. In
Fig. 5
, we have summarized these equations into an algorithmic block diagram. In the next section, we present an example that shows the validity of this algorithm.
Block diagram for the reconstruction of the defectfree mask signal.
IV. EXAMPLE
In this section, we demonstrate the validity of the equations with an example.
 4.1. Precalculation
We assume that
N
= 32 and the defectfree mask and the defected mask produce 32point output signals as follows:
From this, the defect is given as a signal
which, using the delta notation as in Eq. (2), can be represented as
Here,
n
_{1}
= 8,
A
_{1}
= 7,
n
_{2}
= 9,
A
_{2}
= 5,
n
_{3}
= 10,
A
_{3}
= 3. The graphs of
x
(
n
),
y
(
n
), and
h
(
n
) are shown in
Fig. 6
.
Graphs of the defectfree original signal and a defected signal. The defect is marked as *.
Using these values, we calculate the values of the diffraction patterns of the two mask patterns
r_{x}
(
n
) and
r_{y}
(
n
) as well as their difference
r_{xy}
(
n
) =
r_{x}
(
n
) 
r_{y}
(
n
). These values are shown in
Table 1
and, in particular, the graph of
r_{xy}
(
n
) is shown in
Fig. 7
. Assuming that these values are the only available information, we are going to determine the location and values of the defect and the defectfree mask pattern.
Values of the autocorrelation functions and its difference
Values of the autocorrelation functions and its difference
Plot of r_{xy}(n)
 4.2. Determination of the Boundary Values
From
Table 1
and Eq. (8), we determine the boundary values from
r_{x}
(
N
1) =
x
(0)
x
(
N
1) = 100;
 4.3. Determination of the Defect Signalh(n)
As can be seen in
Table 1
as well as in
Fig. 7
, the support of
r_{xy}
(
n
) is given as [23, 23],
n
_{1}
is given as
N
1
n
_{1}
= 31
n
_{1}
= 23, i.e.,
n
_{1}
= 8. And
A
_{1}
can be obtained from Eq. (10)
To determine the size and location of the next defect points, we need to calculate
r
_{xy_temp,1}
(
n
). In
Fig. 8
, we plot the
r_{xy}
(
n
) and
r_{xy_temp,i}
(
n
), for
I
= 1,2,3.
Plot of r_{xy}(n) and r_{xy_temp,i}(n) at each iteration.
As we can see in
Fig. 8(b)
, the boundary value is at
n
= 22. This should be
n
_{2}
= 3122 = 9. If we do the same with this point using the same Eq. (10), then we get
A
_{2}
:
Similarly we get
r
_{xy_temp,3}
(
n
) as in
Fig. 8(c)
and
n
_{3}
= 3121 10 and
A
_{3}
:
If we calculate
r
_{xy_temp,4}
(
n
) as is shown in
Fig. 8(d)
, the signal values are all below 0 and we can terminate this process. Collecting these, we can estimate the defect signal as
From this, we can calculate its autocorrelation
With this, we can calculate
r_{xyh}
(
n
) =
r_{xy}
(
n
) +
r_{h}
(
n
) which is shown in
Table 2
.
Values of the signalsrxy(n),rh(n),rxyh(n)
Values of the signals r_{xy}(n), r_{h}(n), r_{xyh}(n)
Values of the mask pattern in Region #1
Values of the mask pattern in Region #1
Values of the mask pattern in Region #2
Values of the mask pattern in Region #2
 4.4. Reconstruction of the Defectfree Mask Signalx(n)
The mask signal can be reconstructed as follows:
 4.4.1. Region #1
In Region #1, the signal can be reconstructed using Eq. (13).
 4.4.2. Region #2
The values of mask pattern in Region #2 can be obtained from the definition of the autocorrelation function of the mask signal
x
(
n
), i.e., Eq. (14). Since
n
_{1}
= 8 and
n_{M}
=
n
_{3}
= 10, the boundary of Region #1 is
. These values in this area are obtained as follows:
 4.4.3. Region #3
The values of the mask pattern in this region can be obtained by using Eq. (15).
Collecting all these values, we can reconstruct the defectfree mask pattern and is shown in
Fig. 9
. In this figure, we also compared the original defectfree mask signal and the reconstructed signal. The last figure shows the error signal between the original signal and the reconstructed signal. If we define the mean square error as
Comparison of the two signals. (a) Defectfree mask signal, (b) Reconstructed mask signal, and (c) Error signal.
then the MSE is given 1.6916×10
^{27}
in this case and the difference between the original signal and the reconstructed signal is less than 1×10
^{13}
, which is not significant considering calculation errors.
In
Fig. 10
, we have presented an error analysis with respect to the measurement noises. The xaxis and yaxis show the variances of noises that are added to the measured diffraction pattern values and the mean squared error in dB calculated using Eq. (16), respectively. As we can see in this figure, if noises are inserted, it influences the performance of the algorithm. This is true for the many closedform phase retrieval algorithms
[15]
. However, in this case we can see that the effect of the measurement error is not much dependent on the noise level if the noiselevel is below a certain level.
Variation of meansquarederror values with respect to the measurement noises of Fourier transform magnitudes.
Values of the mask pattern in Region #3
Values of the mask pattern in Region #3
V. CONCLUSION
In this paper, we considered a method for the estimation of the location and the amount of stripetype defect in an L/S type mask and the reconstruction of a defectfree mask in lithography. First, we derived an algorithm to estimate the location and depth of the defect signal. Secondly, we derived a deterministic algorithm for the reconstruction of the defectfree mask signal. Using the estimated defect signal and defectfree mask signal, we could also estimate the mask signal with the defect. We present an example that proves that the algorithm works.
It is known that the EUVL is important in developing nanoscale semiconductor chips. The main contribution of this paper is that if we have a defectfree as well as a slightly defected mask pattern, then by measuring the diffraction patterns of these two mask patterns, not only we can estimate the shape and the location of the defect but also determine the exact shape of the defectfree mask pattern. Also, we can use this method for comparing two mask patterns; not only we can find the difference but also we can reconstruct the exact shapes of the two mask patterns, which may be used in the automatic detection /classification of defected masks.
Acknowledgements
Basic Science Research Program of National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (NRF2013R1A1A2010664).
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