Improved ptychographic inspection of EUV reticles via inclusion of prior information

Paolo Ansuinelli; Wim M. J. Coene; Wim M. J. Coene; H. Paul Urbach

doi:10.1364/AO.395446

1. INTRODUCTION

Since extreme ultraviolet (EUV) lithography is nowadays adopted for high volume manufacturing (HVM) in the semiconductor industry, it becomes relevant to address various technical and technological challenges on its roadmap [1]. One of the crucial concerns is mask defectivity [2]. While the technology matures and the technical advancements pave the way toward the printability of increasingly fine features, defects of smaller sizes may also become printable. This dictates the need for highly sensitive mask inspection tools that can generate highly resolved defect maps for defect review and inspection. The EUV mask can be inspected at a nonactinic wavelength [3]; however, the images obtained by means of deep ultraviolet (DUV) and e-beam tools can differ significantly from the aerial images of the scanner [4,5]. This, in turn, can result in the incapability to detect crucial defects that can lead to device failure. Hence, the actinic inspection of EUV masks is particularly important. Further, as mask manufacturers are considering phase shifting EUV mask absorbers for the next generation of EUV technology, methods for metrology and quantification of these phase shifts are important.

Ptychography [6]—a lensless imaging method that enables wide field-of-view, high resolution imaging via phase retrieval—is a possible candidate for the inspection of samples, in reflective mode at short wavelengths [7–9]. This coherent diffractive imaging (CDI) method has been introduced as a potential actinic inspection tool for EUV mask inspection [10–12]. In ptychography, a probe sequentially illuminates a given scattering object at partially overlapping positions. The scattered light is usually detected in the far-field, and the recorded diffraction patterns are computationally processed to image the transmission/reflection function of the scattering object. The translational diversity and the highly redundant information in the dataset result in a robust solution of the phase problem. Nevertheless, since the phase problem is a nonlinear and nonconvex inverse problem, ill-defined solutions can still arise, and there are neither theoretical guarantees on the uniqueness of the retrieved solution, nor certainties about its optimality [13]. In such cases, including prior information in the optimization algorithm can be beneficial.

The use of prior information is ubiquitous in the inverse problems community [14,15], and it was proven to be important to unveil deep subwavelength details in optical imaging methods [16–18]. In the context of CDI, prior information about the amplitude of the ptychographic illumination function enabled subwavelength imaging at the edges of the lines of a periodic object [19]. Recent studies have further shown the benefits that stem from the inclusion of the total variation (TV) prior in the ptychographic algorithms [20–22].

In this work, we have taken a different approach, and instead of using a prior for the probe or a TV prior, we have devised a way to generate a physically sound prior for the transmission/reflection function of the scattering object.

In what follows, we show how one can make use of rigorous forward modeling with Maxwell solvers to compute such prior and how this can be included in the ptychographic method. We apply the algorithm to the problem of detection and imaging of extrusion and intrusion-type defects in a patterned EUV mask layout. The method is shown to outperform the regular ptychographic iterative engine (PIE) and the TV-based ptychographic method.

2. METHOD

Ptychography can be framed as a cost functional minimization problem in which, considering the $j$th probe position $P({\boldsymbol r} - {{\boldsymbol R}_j})$, one seeks a certain object ${O_j}({\boldsymbol r})$ that best fits the $j$th recorded diffraction intensities ${I_j}({\boldsymbol k})$,

(1)$$\begin{split}{{\cal L}({O_j},{I_j})} &= :{{{ {\left|\left| {|{\cal F}(P({\boldsymbol r} - {{\boldsymbol R}_j}){O_j}({\boldsymbol r}))| - \sqrt {{I_j}({\boldsymbol k})}} \right|\right|}}^2}}\\{{O_j}({\boldsymbol r})} &= :\mathop{\arg\min}\limits_{{O_j}}\,{\cal L}({O_j},{I_j}),\end{split}$$

where ${\cal F}$ represents the Fourier transform, $P({\boldsymbol r})$ is the probe function, ${O_j}(r)$ is the part of the guessed object that is illuminated when the probe is located at the $j$th probe position, and ${I_j}({\boldsymbol k})$ is the intensity pattern measured at the $j$th probe position. In Eq. (1) and in all that follows, we assume the probe to be known. The PIE proceeds in a sequential fashion and the complete object, $O({\boldsymbol r})$, is reconstructed by the sequential reconstruction of its $j$th views [6]. Owing to the translational diversity and to the redundancy in the data, the ptychographic reconstructions are very robust. However, as the phase problem is a nonconvex optimization problem—meaning that there can be multiple local minima—and because the reconstruction is obtained solely by processing the dataset, one might wonder whether the inclusion of physically sound prior knowledge in the algorithm can lead to a more satisfactory solution. Prior information is commonly included introducing a regularization term to the data discrepancy functional. In this case,

(2)$$\begin{split}{O_j^\alpha (r)} &= :{{\rm arg}\, {\rm min}\{{\cal L}({O_j},{I_j}) + {R^\alpha}({O_j})\}}\\& = :{{\rm arg}\, {\rm min}\{{{\cal L}^\alpha}({O_j},{I_j})\}},\end{split}$$

where ${R^\alpha}({O_j})$ is the regularizer, which depends upon the regularization parameter $\alpha$, ${{\cal L}^\alpha}({O_j},{I_j})$ is the regularized cost functional, and $O_j^\alpha (r)$ is the associated object update. ${R^\alpha}({O_j})$ is intended as a penalty term in variational approaches or as the negative logarithm of a prior probability distribution from the Bayesian viewpoint [23]. Different regularizers (priors) have been studied in the field of image reconstruction, for instance, structural priors [24], sparsity promoting priors [25], or edge-preserving priors. An important example of the latter is TV, in which ${R^\alpha}(O) =\; \parallel\! \nabla O({\boldsymbol r}\!{)\parallel_1}$. TV is commonly employed due to its effectiveness in smoothing noise—by favoring images that have a sparse gradient—while preserving edges. In recent years, there have been a number of studies reporting implementations of TV regularized ptychography [20–22]. In this work, we have taken a different approach, and instead of considering a regularizer that promotes generic properties of the reconstructed image (like its sparsity or the sparsity of its gradient), the regularizer ${R^\alpha}(O)$ has been designed to promote adhesion to a given prior image (object),

(3)$${R^\alpha}({O_j}) = \alpha { {\parallel\! {{O_j}({\boldsymbol r}) - {O_{p,j}}({\boldsymbol r})} \!\parallel}^2},$$

where ${O_{p,j}}({\boldsymbol r})$ is the prior of the object at the $j$th probe position. Crafting a “good” prior is challenging. This task requires one to properly account for the physics that contribute to the process of image formation. For instance, it is known that at EUV wavelength waveguiding effects and diffraction of light along the thickness of the absorber materials have an impact on image formation [26]. Further, the phase of the scattered wave $\Phi = {\boldsymbol k} \cdot {\boldsymbol r}$ accumulates linearly as photons pass through an object, and the rate of accumulation depends on the refractive index, which changes from material to material [27]. This is important to realize when one wants to compute an a priori map for the phase of a wave scattered by nanostructures that consist of a sequence of layers of different materials and thicknesses. Such a situation applies to the multilayer Bragg reflector and to the absorber layers of an EUV mask. These aspects should be properly encoded in the prior, in order for it to be reliable and accurate.

Fig. 1. Schematic that illustrates the approach discussed in this work. In the detection path—highlighted in blue—the actual defective mask is illuminated by the probe, and the diffracted light is measured in the far-field. In the simulation path—highlighted in yellow—the probe illuminates the nominal defect-free mask, and the far-field is computed, in amplitude and phase, via the forward Maxwell solver. The computationally generated far-field is interpreted in terms of the 2D ptychographic approximation of light–matter interaction [Eq. (4)] and contains information about the prior object ${O_{p,j}}({\boldsymbol r})$. The simulated and detected far-field patterns are subsequently employed in the reconstruction algorithm and in the update rule [Eq. (9)].

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These physical aspects and the intrinsically three-dimensional (3D) thickness effects can be duly accounted for recurring to 3D fully rigorous simulations. Forward Maxwell solvers can compute the complex field, which is a rigorous solution in terms of amplitude and phase of Maxwell’s equations, for a given 3D scattering geometry [28,29]. It is important to notice here that light–matter interaction is modeled differently in the rigorous Maxwell solvers and in ptychography. On the one hand, the rigorous electromagnetic solvers provide an accurate evaluation of light scattering by solving Maxwell’s equation; on the other hand, ptychography models light–matter interaction in terms of the two-dimensional (2D) “probe–times–object” approximation. This fundamental difference in the physical modeling could be a reason of concern when intermixing the use of the rigorous solvers with ptychographic algorithms. In other words, a certain rigorously computed complex-valued far-field, ${\Psi ^{\text{Maxw}}}$, can be used in ptychographic algorithms only when it can be interpreted in terms of the 2D ptychographic approximation of light–matter interaction,

(4)$$\Psi _{p,j}^{\text{Maxw}} \approx {\cal F}(P({\boldsymbol r} - {{\boldsymbol R}_j}){O_{p,j}}({\boldsymbol r})),$$

where $\Psi _{p,j}^{\text{Maxw}}$ is the far-field, amplitude and phase, as computed by the forward Maxwell solver, for the nominal—a priori known—scattering geometry. When Eq. (4) holds, it is possible for us to retrieve the prior for the transmission/reflection function of the object from the rigorously computed far-field, ${O_{p,j}} = P_j^*\frac{{{{\cal F}^{- 1}}(\Psi _{p,j}^{\text{Maxw}})}}{{|{P_j}{|^2}}}$. Notice that although ptychography assumes the object function to be 2D, $\Psi _{p,j}^{\text{Maxw}}$ is computed by the rigorous 3D simulations. In other words, the 2D object reflection function comprises genuine 3D information about the object when computed from $\Psi _{p,j}^{\text{Maxw}}$. Bearing Eq. (4) in mind, we proceed to write, at the $j$th probe position, the ptychographic cost functional inclusive of the prior term. By Eqs. (1)–(3),

(5)$$\begin{split}{{{\cal L}^\alpha}({O_j},{I_j})} &= :{{\cal L}({O_j},{I_j}) + {R^\alpha}({O_j})} \\[-6pt] &={ {{\left| {\left| {|{\cal F}(P({\boldsymbol r} - {{\boldsymbol R}_j}){O_j}({\boldsymbol r}))| - \sqrt {{I_j}({\boldsymbol k})}} \right|} \right|}^2}}\\[-6pt]& \quad+ {\alpha {{\left| {\left| {{O_j}({\boldsymbol r}) - {O_{p,j}}({\boldsymbol r})} \right|} \right|}^2}},\end{split}$$

where $\alpha$ regulates the interplay among the two terms on the right-hand side of Eq. (5). The second term in Eq. (5) penalizes large deviations of the reconstructed object from the prior object. Eq. (5) can be minimized analytically using Wirtinger calculus to yield the update rule that can be used in a gradient descent algorithm. The gradient of Eq. (5) with respect to $O_j^*$ is

(6)$$\begin{split}{\nabla {{\cal L}^\alpha}({O_j},{I_j})} &= P_j^*\left[{{{\cal F}^{- 1}}({\Psi _j} - {\Psi _{c,j}})} \right] + \alpha ({O_j} - {O_{p,j}})\\[-5pt]& = {{\nabla {\cal L}_1} + \nabla {\cal L}_2^\alpha}.\\[-1.5pc]\end{split}$$

In Eq. (6) ${\Psi _j} = {\cal F}({P_j}{O_j})$ is the guessed wavefront in the momentum space, ${P_j} = P({\boldsymbol r} - {{\boldsymbol R}_j})$, and ${\Psi _{c,j}}$ is the revised wavefront obtained by enforcing the measured amplitudes on the far-field guess. We focus now on $\nabla {\cal L}_2^\alpha$, and, with the use of Eq. (4), we obtain

(7)$$\begin{split}{\nabla {\cal L}_2^\alpha }&= {\alpha ({O_j} - {O_{p,j}}) = \alpha \frac{{P_j^*}}{{(|{P_j}{|^2})}}({P_j}{O_j} - {P_j}{O_{p,j}}) }\\[-5pt]&={ \alpha \frac{{P_j^*}}{{|{P_j}{|^2}}}{{\cal F}^{- 1}}({\cal F}({P_j}{O_j}) - {\cal F}({P_j}{O_{p,j}}))}\\[-5pt]& = {\alpha \frac{{P_j^*}}{{|{P_j}{|^2}}}{{\cal F}^{- 1}}({\Psi _j} - \Psi _{p,j}^{\text{Mawx}})}.\end{split}$$

The steepest descent update, at given iteration $n$, dictates

(8)$${O_{j,n + 1}} = {O_{j,n}} - \beta \nabla {{\cal L}^a}.$$

Therefore, the part of the object that is illuminated by the probe at the $j$th probe position, ${O_j}({\boldsymbol r})$, is updated as follows:

(9)$$\begin{split}{{O_{j,n + 1}}} &= {{O_{j,n}} + \beta \frac{{|{P_j}|}}{{|{P_{j,\text{max}}}|}}\frac{{P_j^*}}{{(|{P_j}{|^2} + c)}}({\Psi _{c,j,n}} - {\Psi _{j,n}})}\\&\quad+{ \alpha ^\prime \frac{{|{P_j}|}}{{|{P_{j,\text{max}}}|}}\frac{{P_j^*}}{{(|{P_j}{|^2} + c)}}(\Psi _{p,j}^{\text{Maxw}} - {\Psi _{j,n}})}.\end{split}$$

In Eq. (9), $c$ is a small constant that prevents division by zero, $\alpha ^\prime = \beta \alpha$, and $|{P_j}|/|{P_{j,\max}}|$ is a scaling factor—present in the PIE—that makes the update less reliable where the probe is dim. Although the prior ${O_{p,j}}$ does not appear explicitly in Eq. (9), it is contained implicitly in its second term as ${O_{p,j}} = P_j^*\frac{{{{\cal F}^{- 1}}(\Psi _{p,j}^{\text{Maxw}})}}{{|{P_j}{|^2}}}$. Notice that, besides $\alpha ^\prime $ and $\beta$, the scaling factors in Eq. (9) are the same for the two terms. This is in order to avoid overenforcing the prior in those pixels where the probe is dim. For $\alpha = 0$, Eq. (9) yields the same update rule of the PIE. The algorithm proceeds in a sequential fashion, and the complete object, $O({\textbf r})$, is reconstructed by the sequential reconstruction of its $j$th views, according to Eq. (9).

A layout of this approach is given in Fig. 1.

The inclusion of prior information as a regularizer in Eq. (5) is preferable to the use of ${O_p}({\boldsymbol r})$ as a starting guess in the optimization. There are various reasons for this:

• The presence of the quadratic term in Eq. (5) improves the conditioning of the problem. This stabilizes the inversion with respect to the noise and improves the performance of the iterative method. Further, the regularizer aids in creating a better model by achieving a balance in the bias-variance trade-off for a proper selection of $\alpha$ [30]. Values of $\alpha$ that are too small make the reconstruction too sensitive to the noise; however, setting $\alpha$ to a value that is too high biases the reconstruction toward the prior, yielding a poor fit. A proper value of $\alpha$ achieves a balance among these two cases.
• The prior in Eq. (9) is present at every step of the iteration, therefore preventing the reconstruction from diverging toward an “unphysical” solution, and
• the regularization parameter $\alpha$ can be tuned to reflect the degree of belief in the prior.

In what follows, we will show that the inclusion of the regularization term, Eq. (3), in the cost functional Eq. (5) via the use of the accurate physical–analytical models yields a better reconstruction with respect to the standard case in which the reflection function $O({\boldsymbol r})$ is retrieved solely by processing the intensity dataset as in Eq. (1).

Table 1. Layers Thicknesses and Materials at $\lambda = 13.5\;{\rm nm} $

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3. RESULTS

We have applied the method outlined above to reconstruct the patterned absorber of a 3D EUV mask. Table 1 reports the materials and the thicknesses of the layers used in this work.

Four EUV masks have been considered through this study:

• The “nominal” mask. This is the cell as given by prior information. This cell does not contain any information about the defects. This is the cell used in the simulation path depicted in yellow in Fig. 1.
• The defect-free “actual” mask. This is the cell that mimics the “reference” mask, which is close to the prior but not exactly the same. In order to account for this difference, the actual cell has been generated from the prior cell, displacing the sides of the polygons of 1–5 nm. This cell is displayed in Fig. 2(a).
• The “actual defective” or programmed defects mask. Consistently with the practice in EUV mask defectivity studies, we have perturbed the actual mask, at known locations, with additive and subtractive features (extrusions and intrusion defects) [Figs. 2(b) and 2(c)]. This is the cell used in the detection path depicted in blue in Fig. 1.

The size of the defects in Figs. 2(b) and 2(c) is the same on a given polygon, and it changes from polygon to polygon. The number and the side length of the squares that constitute the rough extrusions/intrusions on a certain polygon are the following: [number of squares, side length] = [3, 16 nm], [6, 12 nm], [7, 9 nm], [7, 6 nm]. Such sizes have been chosen in accordance to the theoretical Abbe resolution limit imposed by the NA (11 nm in this case). If the collection NA was to be smaller, the defects would have been made bigger accordingly.

Fig. 2. Top views of EUV mask layouts. (a) Actual cell, (b) programmed defect mask (extrusions), (c) programmed defect mask (intrusions). The side length of the single defect is specified in the figure.

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To understand whether the inclusion of prior information yields any benefits for our specific application, we have carried out a computational die-to-database comparison [31]. This is done comparing the reconstruction of the defective mask with the reconstruction of the defect-free actual mask. The two reconstructions are subtracted one from the other to identify the defects at their locations. The impact of the defects is quantified by a certain figure of merit. In what follows, we will use the defect signal-to-noise (SNR) defined as [31]

(10)$${{\rm SNR}_\delta} = \frac{{{{\bar A}_d} - {{\bar A}_a}}}{{\text{std}({A_a})}},$$

where ${\bar A_d}$ is the average magnitude of the defected area, ${\bar A_a}$ is the average magnitude of the whole difference image—where the object is present—and ${\rm std}({A_a})$ is the standard deviation of the latter area. The definition of the defect SNR is independent of the defect size. This investigation is done using the standard PIE and the PIE with prior, where we use Eq. (9) as update rule to reconstruct both the actual mask [Fig. 2(a)] and the defected masks [Figs. 2(b) and 2(c)].

As outlined above, four datasets have been computed:

• One complex dataset, in amplitude and phase, for the nominal mask. This is the far-field $\Psi _{p,j}^{\text{Maxw}}$, related to the nominal mask we referred to in Eqs. (4) and (9).
• One intensity-only dataset for the actual defect-free mask in Fig. 2(a). This dataset has been corrupted with noise to emulate measured data.
• Two intensity-only datasets for the defected masks, one for the extrusions in Fig. 2(b) and one for the intrusions in Fig. 2(c). These datasets have also been corrupted with noise to emulated measured data.

Fig. 3. (a) Probe used in the computations. (b) Cross section of (a). (c) Diffraction pattern (log scale).

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Fig. 4. Zoom of the prior object: (a) magnitude, (b) phase, and (c) magnitude after and removal of the strong unscattered beam to emulate the presence of a beam-stop in the experimental setup. The fringes in (b) are caused by the projection of the term ${e^{i{\boldsymbol k} \cdot {\boldsymbol r}}}$ over the $x {-} y$ plane related to the 6° tilt of the probe with respect to the normal.

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The datasets mentioned above have been generated via fully rigorous 3D simulations using a volume-integral Maxwell solver [32,33]. The solver is formulated for the problem of scattering from periodic objects; hence, in order to avoid cross talk among adjacent cells, we have opted for a supercell approach. The cell is a square with lateral dimension $\Lambda = 3.5\,\,\unicode{x00B5}{\rm m}$. The sampling in the far-field and in the illuminating NA equals $\frac{{2\pi}}{\Lambda}$. Although the lateral dimensions of the supercell are of the order of hundreds times the wavelength, the computational complexity and the memory requirement necessary to solve the forward problem are maintained relatively low, of the order of ${\cal O}(N\log N)$. The probe is assumed to be a Gaussian beam with a $3\sigma$ amplitude of about 15 µm, and it is described by its angular spectrum. The scattered far-field is evaluated, for each of the plane waves that compose the illumination, in parallel on a multicore high performance computing (HPC) cluster. The output far-field that results from the interaction of the probe with the object is then given by the weighed coherent superposition of the individual contributions. The ptychographic scans are performed shifting the object of 0.2 µm, in five positions, inside the supercell. The probe is polarized in the $x$ direction—parallel to the horizontal axis of the supercell—by proper linear combination of $s\!$- and $p$-polarization states. The collection NA is 0.6, close to the value (0.54) used for an identical wavelength in [19]. Figures 3(a) and 3(b) illustrate the probe, its cross section, and one of the acquired diffraction patterns for the mask in Fig. 2(a).

Figures 4(a) and 4(b) shows the prior for the central probe position. In all that follows, we have fixed the regularization parameter $\alpha$ to 2e-2.

A. Extrusion Defects

The magnitude of the ptychographic reconstruction of the patterned absorber depicted in Fig. 2(b), without and with prior respectively, is depicted in Figs. 5(a) and 5(b), while the phase is shown in Figs. 5(d) and 5(e). White Gaussian noise has been added to the synthetically generated data for an ${\rm SNR} = {110}\;{\rm dB}$. The object error, at iteration $n$, has been computed as the deviation of the reconstructed object, ${O_n}({\boldsymbol r})$, from the theoretical object ${O_t}({\boldsymbol r})$,

(11)$${E_{O,n}} = \frac{{\sum_{\textbf r} |{O_t}({\textbf r}) - \gamma {O_n}({\textbf r}{{)|}^2}}}{{\sum_{\textbf r} |{O_t}({\textbf r}{{)|}^2}}},$$

where $\gamma$ is a parameter that compensates for phase ambiguities [34].

Fig. 5. Ptychographic reconstructions: (a) magnitude as given by the PIE; (b) magnitude given by the new algorithm PIE with prior term included. The rough extrusions are highlighted in the red circles; (c) error; (d) phase given by the PIE, and (e) phase given by the PIE with prior.

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Result of the reconstructions is highlighted in Figs. 5(a)–5(d). The rough extrusions, highlighted in Fig. 5(b), are better resolved in both the amplitude and phase images in Figs. 5(b) and 5(d). The error, defined in Eq. (11) and shown in Fig. 5(c), shows an overall better reconstruction and convergence when the prior is included in the optimization algorithm. Practically, we have observed the algorithm to converge in about a third of the iterations of the regular PIE. As stated before, a quantitative assessment of the improvement in terms of defect inspection can be obtained in a die-to-database comparison by computing difference images. Those are defined as the magnitude of the difference among the retrieved ptychographic reconstruction of the objects in Figs. 2(a) and 2(b),

(12)$$\Delta\! O({\textbf r}) = |{O_a}({\textbf r}) - {O_\delta}({\textbf r})|,$$

with ${O_a}({\textbf r})$ and ${O_\delta}({\boldsymbol r})$ being the reconstructed reflection functions relative to actual mask in Fig. 2(a) and to the defected mask in Figs. 2(b) and 2(c). In what follows, we will refer to Eq. (12) as the object difference metric. Figure 6 shows $\Delta O({\textbf r})$ defined in Eq. (12), obtained when reconstructing ${O_a}({\boldsymbol r})$ and ${O_\delta}({\boldsymbol r})$ using prior information—by the ptychographic update rule (9)—and the standard PIE. Here, and in all that follows, the objects have been aligned before their subtraction.

Fig. 6. (a) and (b) show the object difference metric $\Delta\! O$ in Eq. 12 obtained for extrusions. (a) makes use of the prior while in (b) we employed the standard PIE. The sizes in (a) are the side lengths of each of the blocks of the extrusions.

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All the defects are better resolved in Fig. 6(a) than in Fig. 6(b), and their signature appears to be stronger in the difference image Fig. 6(a). Particularly, the rough defects of 9 nm size are not distinguishable in Fig. 6(b); however, they are detectable in Fig. 6(a), as highlighted in the red circle. We found the finest details, the rough extrusions of 6 nm size, to be absent in the reconstruction in Figs. 5(a) and 5(b), and in the difference images in Figs. 6(a) and 6(b). The side of the polygon over which these defects are located appears to be smooth in the reconstructed image. This computational experiment highlights that subwavelength ptychographic imaging is possible to a certain extent; however, the spatial periodicities below the theoretical limit of about $\lambda /2$ are not retained, in this case, in the reconstruction. Being the collection NA equal to 0.6, the Abbe limit is 11 nm, which is slightly above the size, 9 nm, of the smallest defect we managed to resolve. A comparison of the retrieved SNR for Figs. 6(a) and 6(b) is reported in Table 2.

Table 2. Extrusion Defects SNR

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Table 2 highlights a steep improvement in the detectability of defects when incorporating the prior term in the reconstruction algorithm. The value N/A means that the defects are not visible in the difference image. A cross section of the reconstructions in Figs. 5(a) and 5(b), taken on the location of the defects, is shown in Fig. 7.

Fig. 7. Cross section of the reconstruction at the defect sites. (a) Three defects of 16 nm side length; (b) six defects of 12 nm side length; (c) seven defects of 9 nm side length.

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Fig. 8. Ptychographic reconstructions: (a) magnitude as given by the PIE; (b) magnitude given by the PIE with prior included. The rough inclusions are highlighted in the red circles; (c) phase given by the PIE, and (d) phase given by the PIE with prior; (e) error.

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The cross sections in Fig. 7 generally have a move pronounced peak-to-valley ratio when the prior is included, and, in the case of Fig. 7(c), the periodicity of the signal is more evident, while it is lost—in the central part of the plot—in the case of the standard PIE.

B. Intrusions

The magnitude of the ptychographic reconstruction of the patterned absorber layout in Fig. 2(c), without and with prior, respectively, is depicted in Figs. 8(a) and 8(b), while the phase is shown in Figs. 8(d) and 8(e).

Figure 9 shows the object difference metric $\Delta O({\textbf r})$ defined in Eq. (12), obtained using the update rule Eq. (9) and the standard PIE.

The comparison of the SNR for Figs. 8(a) and 8(b) is reported in Table 3.

Fig. 9. (a) and (b) show $\Delta O$ in Eq. (12) obtained for intrusions. (a) makes use of the prior while in (b) we employed the standard PIE. The sizes in (a) are the side lengths of each of the blocks of the intrusions.

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All the intrusion defects have decreased SNR with respect to extrusion defects of the same size, and even with the inclusion of the prior term, the 9 nm size intrusions are difficult to image. As guided modes propagate within the absorber, and as they are not immediately truncated at the edges of the structures, they can couple and interfere within the small intrusions, making these defects harder to image with high contrast [26]. Figure 10 illustrates the cross section of the reconstructions in Figs. 8(a) and 8(b), taken at the location of the defects.

Table 3. Intrusion Defects SNR

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Fig. 10. Cross section of the reconstruction at the defect sites. (a) Three defects of 16 nm side length; (b) six defects of 12 nm side length; (c) seven defects of 9 nm side length.

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Fig. 11. Ptychographic reconstructions: (a) magnitude as given by the PIE; (b) difference image given by PIE; (c) error.

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C. Effect of Number of Probe Positions, SNR, Initial Guess and a Comparison with TV Regularization

1. Increasing the Number of Probe Positions

Ptychography achieves a very robust reconstruction by exploiting translational diversity and redundancy in the dataset. In the study presented earlier, the dataset included five probe positions. As the object is, in this case, entirely covered by the probe, there is a high degree of redundancy in the data in spite of the few probe positions. However, it can be interesting to see whether increasing the number of probe positions allows one to get to a reconstruction, which is as good as in the case in which the prior term is included. We have performed this study for the case of the extrusion-type defects. We have used nine probe positions, and the reconstruction has been carried out using the PIE. The nine positions constitute a 3-by-3 grid which span, in steps of 200 nm along the $x$ and $y$ directions, a square whose side length is 400 nm. The reconstruction and the difference image are shown in Fig. 11.

Figure 11(b) reports an overall better reconstruction of the defects with respect to Fig. 6(b). Still the reconstruction is not as satisfactory as the one obtained with five probe positions and the inclusion of the prior term. This is highlighted in Table 4.

2. Decreasing SNR

The effect of the noise is here studied. A decrease of the SNR constitutes a problem for the retrieval of fine features that weakly scatter light, because their signature could be below the noise floor. A workaround could be to increase the radiation dose, but this could in turn damage the sample. The role of the prior as a regularizer is helpful in this, as it stabilizes the inversion and enables one to achieve a better reconstruction when the SNR decreases. Here we have decreased the SNR from 110 dB to 100 dB and 90 dB. The results, for the extrusion defects, are given in Figs. 12 and 13.

All of the defects are still visible in Fig. 12(a), while in Fig. 13(b), only the bigger ones are visible. In Fig. 13, the 16 nm and 12 nm size defects are visible, but none of the defects can be detected by the PIE.

3. Using the Prior as Initial Guess

The advantages of having a regularizer in Eq. (5) rather than using the prior only as a starting point were outlined in Section 2. Here we compare the results, for extrusion-type defects, given using the update rule Eq. (9), the PIE, and the PIE using the prior as starting guess. Figure 14 highlights results for an ${\rm SNR} = {110}\;{\rm dB}$. Figure 15 shows results for an ${\rm SNR} = {100}\;{\rm dB}$.

As can be noticed, a proper starting guess in the standard PIE yields a better reconstruction; however, the absence of the regularizer—that stabilizes the reconstruction with respect to the noise and that promotes the retrieval of a better fit via a bias-variance trade-off—has a negative impact over the reconstruction.

Table 4. Extrusions Defects SNR

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Fig. 12. (a) and (b) show the object difference metric $\Delta O$ in Eq. (12) obtained when ${\rm SNR} = {100}\;{\rm dB}$. (a) PIE with prior; (b) PIE.

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Fig. 13. (a) and (b) show the object difference metric $\Delta O$ in Eq. (12) obtained when ${\rm SNR} = {90}\;{\rm dB}$. (a) PIE with prior; (b) PIE.

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Fig. 14. Difference image SNR 110 dB: (a) update rule Eq. (9); (b) PIE; (c) PIE with prior as starting guess.

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Fig. 15. Difference image SNR 100 dB: (a) update rule Eq. (9); (b) PIE; (c) PIE with prior as starting guess.

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Fig. 16. Object difference image SNR 110 dB: (a) update rule (9); (b) TV regularization; (c) error.

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Fig. 17. Object difference image SNR 90 dB: (a) update rule (9); (b) TV regularization; (c) error.

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4. Comparison with TV Regularization

Recent works have reported the use of TV as regularizer for denoising in ptychography [20–22]. Here we compare results obtained using the two different regularizers ${R^\alpha}({O_j}) = \alpha ||{O_j}({\boldsymbol r}) - {O_{p,j}}({\boldsymbol r}{)||^2}$ and ${R^{{\alpha _\textit{TV}}}}({O_j}) = {\alpha _\textit{TV}}||\nabla {O_j}({\boldsymbol r}{)||_1}$. Notice that since ${R^{{\alpha _\textit{TV}}}}({O_j})$ is nonsmooth, gradient-based approaches are not immediately applicable anymore. One workaround is to replace the ${\ell _1}$ norm by a smooth approximation; another one, employed here, is to solve TV via the alternating direction method of multipliers (ADMM). We begin by writing the following problem at the $j$th probe position:

(13)$$\begin{split}&\!\!\!\!\!\!{\mathop {\min}\limits_{{O_{j,p}}}\, {{\left| {\left| {|{\cal F}(P({\boldsymbol r} - {{\boldsymbol R}_j}){O_j}({\boldsymbol r}))| - \sqrt {{I_j}({\boldsymbol k})}} \right|} \right|}^2} + {\alpha _\textit{TV}}||p{{||}_1}}\\&\quad{{\rm s}{\rm .t}{\rm .}\,p = \nabla {O_j}}.\end{split}$$

The augmented Lagrangian for real functional of complex variables can be written as [35]

(14)$$\begin{split}{{{\cal L}_\rho}({O_j},p,y)}& = {{{\left| {\left| {|{\cal F}(P({\boldsymbol r} - {{\boldsymbol R}_j}){O_j}({\boldsymbol r}))| - \sqrt {{I_j}({\boldsymbol k})}} \right|} \right|}^2}}\\&\quad+{ 2{\rm Re}(\langle p - \nabla {O_j},y\rangle) + \rho ||p - \nabla {O_j}{{||}^2}}\!\!\!\\ &\quad+{ {\alpha _\textit{TV}}||p{{||}_1}},\\[-1pc]\end{split}$$

where $\langle \cdot , \cdot \rangle$ is the inner product over the complex space, ${\rm Re}(z)$ is the real part of $z$, $y$ is the Lagrangian multiplier, and $\rho$ is the penalty parameter. Equation (14) can be further simplified scaling the variable $y$ by $1/\rho$ [36]. At iteration $n$, the ADMM solves the following steps:

(15)$$\begin{split}O_j^{n + 1} & = \mathop{\arg\min}\limits_{{O_j}}\,{{\cal L}_\rho}({O_j},{p^n},{y^n})\\[-3pt] {p^{n + 1}} & = \mathop{\arg\min}\limits_{p}{{\cal L}_\rho}(O_j^{n + 1},p,{y^n})\\[-3pt] {y^{n + 1}} & = {y^n} + \rho ({p^{n + 1}} - \nabla O_j^{n + 1}),\end{split}$$

The derivations necessary to minimize Eq. (15) are analogous to the ones that can be found elsewhere [21] and are here omitted. Equation (13) is minimized per probe position. The subproblem with respect to ${O_j}$ is solved via a steepest descent, the subproblem with respect to $p$ has a closed-form solution in the form of a soft-thresholding operator, and ${\alpha _\textit{TV}}$ has been computed by the L-curve and is equal to 1e-6 [37].

The object difference images for ${\rm SNR} = {110}\;{\rm dB}$ and for ${\rm SNR} = {90}\;{\rm dB}$, for extrusions defects, are outlined in Figs. 16 and 17.

Although TV regularization produces results that are qualitatively superior to the PIE [Figs. 6(b) and 13(b)], the proposed method that includes the object prior in the optimization enables a more robust reconstruction.

4. CONCLUSION

We have presented a ptychographic algorithm that makes use of prior information about the transmission/reflection function of the object to improve the quality of the ptychographic reconstruction. Prior information is generated by employing the rigorous forward Maxwell solver—to properly account for the complex physics that contribute to the process of image formation—and by interpreting the outcome of the fully 3D rigorous simulations in terms of the “probe–times–object” ptychographic approximation of light–matter interaction. The method has been applied to the problem of actinic ptychographic inspection of extrusion- and intrusion-type defects in a patterned EUV mask layout. The numerical results indicate a steep improvement over the standard PIE in terms of quality of reconstruction and convergence. Smaller defects are detectable, and all the defects are imaged with higher SNR. The method is shown to achieve better reconstructions using a smaller number of probe positions, to be more robust to the noise, and to outperform TV-based ptychographic imaging. Although the algorithm has been applied to the specific technological problem of EUV mask inspection, this work can be interesting for the ones that employ lensless imaging for the reconstruction of nanostructures and that have prior information about the object structure available.

Funding

H2020 Marie Skłodowska-Curie Actions (675745).

Acknowledgment

We are grateful to Mark Van Kraaij, ASML Research, for assistance with the use of the Maxwell solver and to the anonymous reviewers whose suggestions helped to improve and clarify this paper.

Disclosures

The authors declare no conflicts of interest.

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Layer	Thickness [nm]	$n$	$k$
ARC TaBO	2	0.952	0.026
Absorber TaBN	58	0.95	0.031
Ru	0.5	0.88586	0.01727
Ru (capping layer)	2	0.88586	0.01727
Si	1.8968	0.99888	0.00183
${M o S i}_{2}$	0.7986	0.96908	0.00435
Mo	2.496	0.92347	0.00649
${M o S i}_{2}$	1.8908	0.96908	0.00435

Defect Size [nm]	Number of Defects	SNR PIE with Prior	SNR PIE
16	3	3.8	1.8
12	6	5.3	1.2
9	7	1.6	N/A

Defect Size [nm]	Number of Defects	SNR with Prior	SNR PIE
16	3	3.8	1.5
12	6	3.4	N/A
9	7	1	N/A

Defect Size [nm]	SNR with Prior (5 Probe Positions)	SNR PIE (9 Probe Positions)
16	3.8	3
12	5.3	2.5
9	1.6	0.6

Layer	Thickness [nm]	$n$	$k$
ARC TaBO	2	0.952	0.026
Absorber TaBN	58	0.95	0.031
Ru	0.5	0.88586	0.01727
Ru (capping layer)	2	0.88586	0.01727
Si	1.8968	0.99888	0.00183
${M o S i}_{2}$	0.7986	0.96908	0.00435
Mo	2.496	0.92347	0.00649
${M o S i}_{2}$	1.8908	0.96908	0.00435

Improved ptychographic inspection of EUV reticles via inclusion of prior information

Abstract

1. INTRODUCTION

2. METHOD

3. RESULTS

A. Extrusion Defects

B. Intrusions

C. Effect of Number of Probe Positions, SNR, Initial Guess and a Comparison with TV Regularization

1. Increasing the Number of Probe Positions

2. Decreasing SNR

3. Using the Prior as Initial Guess

4. Comparison with TV Regularization

4. CONCLUSION

Funding

Acknowledgment

Disclosures

REFERENCES

Cited By

Figures (17)

Tables (4)

Equations (15)

Applied Optics