In this paper, we consider a method for estimating a stripe-type defect and the reconstruction of a defect-free L/S type mask used in lithography. Comparing diffraction patterns of defected and defect-free 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 defect-free mask pattern. An example shows the validity of the equations. 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 input-output (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 off-axis 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 two-dimensional signal irreducible [10] . Podorov et al. and Guizar-Sicario 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 defect-free 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 time-reversal for a one-dimensional signal. The goal is to estimate the location and shape of a stripe defect and distinguish it from the diffraction pattern of a defect-free 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 defect-free 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 defect-mask 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 defect-free 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. We consider a simple stripe-type defect that occurred in an L/S-type 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 defect-free mask and (b) is the cross-section of this mask on the x-y plane. We let the mathematical notation of the defect-free 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 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 cross-sectional 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 defect-free mask has a region of support [0, N -1] and the defect is located on the left side of the mask-all 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 defect-free masks. However, in this case, the estimated defect-free 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 defect-free 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. In this section, we derive equations for the calculation of the values of A_i ’s and n_i ’s. 3.1. Mathematical Preliminaries We assume that we have a prior knowledge of the diffraction pattern of the defect-free 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 defect-free mask pattern or its time-reversed versions, and these two terms are overlapped around the origin. Based on this, we can decompose the support of the defect-free 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. The boundary value x_mask can be determined from the autocorrelation function of the defect-free 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 . Since we do not know the exact shape of the defect-free 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 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. Since in 3.3, we have determined the defect signal, now we can reconstruct the defect-free 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. 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. 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,…, . 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. In this section, we demonstrate the validity of the equations with an example. We assume that N = 32 and the defect-free mask and the defected mask produce 32-point 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 . 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 defect-free mask pattern. From Table 1 and Eq. (8), we determine the boundary values from r_x ( N -1) = x (0) x ( N -1) = 100; 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. As we can see in Fig. 8(b) , the boundary value is at n = 22. This should be n ₂ = 31-22 = 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 = 31-21 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 . The mask signal can be reconstructed as follows: In Region #1, the signal can be reconstructed using Eq. (13). 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: The values of the mask pattern in this region can be obtained by using Eq. (15). Collecting all these values, we can reconstruct the defect-free mask pattern and is shown in Fig. 9 . In this figure, we also compared the original defect-free 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 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 x-axis and y-axis 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 closed-form 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 noise-level is below a certain level. In this paper, we considered a method for the estimation of the location and the amount of stripe-type defect in an L/S type mask and the reconstruction of a defect-free 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 defect-free mask signal. Using the estimated defect signal and defect-free 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 defect-free 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 defect-free 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.