1. INTRODUCTION

In angular-resolved scatterometry, given a certain illumination (incident amplitude, phase, and polarization) and optical system [incident wavelength and numerical aperture (NA)], the set of all elements of the scattering matrix at all angles that can be detected by the system contains the maximum information about the scatter.

In this paper, we explore and demonstrate the limit of optical scatterometry by determining the polarization-resolved amplitude and phase of the scattered field using coherent illumination. The maximum information content not only is relevant for the basic understanding of diffraction problems in optics, but also is crucial for applications where pushing the technique to its limits is required. Important applications are metrology for chip fabrication in the semiconductor industry and surface inspection for particle contamination and defect detection in both semiconductor and plastic electronics.

For example, in optical lithography, along with the ability to print sub-wavelength structures, stringent quality control of the lithographic process is required. Generally, gratings are used in situ as the metrology target. Slight variations in the dose, exposure, or the environmental conditions, such as temperature and pressure, can lead to deviation of the grating shape from the desired one. Measurement of the shape parameters of the gratings can be done in principle with scanning electron microscopes and/or atomic force microscopes, but optical scatterometry is the de-facto preferred method because it is noninvasive and fast.

In optical scatterometry, the retrieval of the shape parameters of the grating is done by matching the experimental far-field scattering intensity distribution with the expected distribution that is calculated by rigorous computations using, for example, the rigorous coupled-wave analysis (RCWA) method [1,2]. To determine the ultimate resolution, one should gather the maximum amount of information about the scattering matrix. There are many variants of optical scatterometry techniques such as single-incidence angle reflectometry, 2-$\mathrm{\Theta }$ scatterometry, spectroscopic ellipsometry, Fourier scatterometry, interferometric Fourier scatterometry, and, in recent years, coherent Fourier scatterometry (CFS) [3–13]. In particular, the latter can be made very fast since a focused coherent beam is used, and the scattering information at many angles is captured in one shot. Furthermore, when the period of the structure is such that diffracted orders overlap and the focused spot is scanned, the phase differences between these orders in CFS makes this technique more sensitive than incoherent Fourier scatterometry [14]. However, diffracted orders only overlap when the pitch is large enough for the given NA of the focusing lens and the illumination wavelength used. This implies that the advantage of CFS over incoherent scatterometry methods is limited to gratings of a certain minimum pitch. Therefore, we have recently proposed [15] an interferometric version of CFS, by which not only the amplitude but also the phase of the scattering matrix elements are determined, and hence higher sensitivity is achieved for arbitrary pitch. In interferometric CFS (ICFS), the polarization-resolved scattered field can be retrieved for not only scattering angles where orders overlap but also any scattering angle. ICFS utilizes a reference wave that interferes with the far field generated by CFS to capture phase information present in the zeroth order and overlapping orders if they exist. In the latter case, the difference in phase between the overlapping orders is retrieved by scanning, as is done for noninterferometric CFS. Furthermore, by measuring all possible orthogonal incident and scattered polarizations, one can recover the entire complex scattering matrix, i.e., all information that can be obtained within the given numerical aperture of the system. Note that we use the same lens to focus the incident wave into a spot and to project the scattered field onto the CCD.

In the present paper, we show experimental proof of the predictions presented in [15] by determining the full scattering matrix using ICFS. The content of the paper is as follows. In Section 2, we summarize the most important aspects of the theory. In Section 3, we describe experimental details, in particular setup and data acquisition. In Section 4, we discuss the comparison with simulations, and Section 5 contains the conclusions.

2. THEORY

Let us consider the scattering problem of a one-dimensional grating. We choose a coordinate system $(x,y,z)$ as shown in Fig. 1 with the $z$ axis coinciding with the optical axis of the focusing system. The permittivity is a periodic function of $x$ and invariant along $y$. It is conventional to set $z=0$ at the top of the grating and to choose $z$ positive in the direction of incidence (see Fig. 1). The incident field is denoted as ${\mathbf{E}}^i$, and the reflected field from the grating is denoted as ${\mathbf{E}}^r$. If the grating thickness is $d$, then we can express the electric fields as ${\mathbf{E}}^i+{\mathbf{E}}^r$ when $z<0$.

 

Fig. 1. Schematic diagram of our approach to the problem of one shot scattering matrix determination for a large number of incident angles. (A) The $x-y-z$ coordinate system is attached to the sample with the $z=0$ plane chosen at the interface between the grating and the incident medium, which is also the geometric focus plane of the objective. (B) The far-field maps of the complex amplitude of the field is projected on the CCD in the exit pupil of the objective where the coordinate system ($\xi -\eta$) is chosen such that $\xi$ and $\eta$ are parallel to the $x$ and $y$ direction, respectively. (C) The interface between the half-space $z<0$ and the grating and the interface between the grating and the substrate $z>d$ are indicated by the dotted lines.

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With Rayleigh’s method [16–18], the total field in the grating region can be computed, for example, with the RCWA [1,19,20]. The reflected field is then obtained by subtracting the incident field from the total field. In the half-space $z<0$, the reflected field is expanded into a sum of plane waves as follows:

In this work, by combining coherent Fourier scatterometry with temporal phase shifting interferometry, we show that it is possible to determine the complex reflection coefficients for many angles of incidence in one shot. The basic idea is to use a microscope objective (MO) to focus incident beam onto the scattering sample, which for our case is the one-dimensional grating (shown in regions A and C of Fig. 1). In region B of the same figure, the far-field coordinates ($\xi -\eta$) are defined by

In practice, because of the presence of the objective, there are some additional geometric transformations to consider. If the objective is perfectly isotropic, it does not create any additional phase difference between the $s$- and $p$-polarized components. In that case, the incident and reflected fields are related through ${\mathcal{R}}_m$ instead of ${\mathbf{R}}_m$, where

In the experimental results that we are going to present, the incident state of polarization is known and the incident field is measured by a wavefront sensor, so that the complex amplitudes in every point of the entrance pupil are known. The incident field is then decomposed on the ($\xi$, $\eta$) basis. The amplitudes of the matrix elements at a particular input/output polarization combination are directly measured by imaging the exit pupil onto a CCD camera, while the phase is obtained interferometrically by combining the field at the exit pupil with a reference beam.

3. EXPERIMENTAL REALIZATION

A. Setup

We designed and built a coherent scatterometer, along with an interferometer functionality, based on temporal phase shifting interferometry. The design is basically the one of a coherent Fourier scatterometer (see Ref. [14]) where a reference mirror with a piezoelectric transducer is added to the open port of the beam splitter (BS). This setup allows us to determine the scattering matrix of any object of interest. In this paper, we consider a periodic silicon-on-silicon square grating described by the parameters listed in Table 1. Those parameters, which have been used as input values for the RCWA simulations, have been measured with atomic force microscopy (grating height and period) and scanning electron microscopy (grating period). A schematic overview is shown in Fig. 2. The light from a He–Ne laser (S) beam ($\lambda =633\mathrm{nm}$) is coupled to a single-mode fiber (SMF). The light coming out from the fiber is collimated (${\mathrm{L}}_1=20\mathrm{cm}$), and the desired input polarization direction is set with a Glan–Taylor polarizer (${\mathrm{POL}}_{\mathrm{in}}$ in the figure). To define the different states of polarization, we will use the notation provided by Section 2. A BS separates the beam for the sample and reference mirror arm. On the sample arm, the polarized light is focused on the grating with a MO (Leitz Wetzlar, $20\times$, infinity corrected) of $\mathrm{NA}=0.4$. In the reference arm, the beam is reflected by a $\lambda /20$ flat aluminum mirror controlled by a piezo translation stage (PZT). The reflected light from the grating in the exit pupil of the MO as seen from the sample is optically conjugated with the detector plane by two lenses (${\mathrm{L}}_2=40\mathrm{cm}$ and ${\mathrm{L}}_3=25\mathrm{cm}$). Before the beam is recorded by the CCD camera, another polarizer (${\mathrm{POL}}_{\text{out}}$) allows the selection of output polarization. In our scheme, output polarization $(\xi ,\eta )$ means that the polarizer is parallel to the input polarization $(\xi ,\eta )$.

Table 1. Physical Dimensions of the Grating Under Investigation

View Table

 

Fig. 2. Schematic overview of the experimental setup. S, He–Ne laser; SMF, single-mode fiber; L1, collimating lens; L2 and L3, telescopic lenses; BS, nonpolarizing beam splitter; ${\mathrm{POL}}_{\mathrm{in}}$ and ${\mathrm{POL}}_{\text{out}}$: polarizers; MO, microscope objective; PZT, piezoelectric translation stage; CCD, data acquisition camera.

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B. Data Acquisition

As previously mentioned, the setup (and therefore its working principle) can be employed to quantify the scattering properties of a variety of different structures. For the etched silicon grating mentioned above and for a wavelength of 633 nm and a $\mathrm{NA}=0.4$ of the MO, only the zeroth order is captured by the CCD camera. Since there are no overlapping orders, the spot does not have to be scanned to retrieve the phase difference between them [14]. We obtain data for four input/output polarization combinations, namely, $\xi -\xi$, $\xi -\eta$, $\eta -\eta$, and $\eta -\xi$. For each of them, intensity frames for several axial positions of the reference mirror are obtained by applying specific voltages to the PZT. The phase of the object is then reconstructed from the intensity data by means of a five-step phase shifting algorithm [23]. In fact, since only the zeroth order is present, only one complex amplitude is required to be calculated, thus the aforementioned algorithm is sufficient. When higher orders are also present, we will need more phase steps.

In the case of phase retrieval by a temporal phase shifting algorithm, there is always an uncertainty in the piezo movement that results in an error in the intended phase change of the reference beam. This error can be minimized either by using phase retrieval algorithms that are less sensitive to the error in the piezo movement and/or by choosing the correct frame corresponding to the intended phase shift (in our case, it is $\pi /2$). We devised a correlation-based technique to minimize the error. The method is explained in Fig. 3. The phase retrieval algorithm is implemented with five intensity patterns recorded for the corresponding $\pi /2$ phase shifted reference arm of the interferometer. The correlation coefficient between the images is computed for displacement versus voltage, which indeed gives the information about the phase shift between the images. Five images are then chosen for the phase retrieval. A single image recorded by the detector is an interference pattern for a defined input and output polarization

 

Fig. 3. Correlation function $\rho$ as a function of the piezo translation stage displacement. The maxima and minimums of the correlation curve are examined to monitor the movement of the piezo. Only four experimental images are shown for simplicity.

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Because we are recording images of an interference pattern, we expect the function ${\rho }_j$ to have a cosine-like behavior (as confirmed by the data trend in Fig. 3). The five different phase steps ${\mathrm{\Phi }}_r$ we need to consider correspond to the maxima and minima of ${\rho }_j$, along with the points of ${\rho }_j=0$ when ${\mathrm{\Phi }}_r=\pi /2,3\pi /2$ over one period. These points are highlighted with red marks in Fig. 3. In this way, by selecting the right images from the experimental data and substituting them into Eq. (8), we can retrieve the phase information we need. The obtained phase data is unwrapped using a quality guided path algorithm [25], specifically the one in Ref. [26]. Experimental data were treated with a smoothing filter to minimize the noise influence using a Gaussian function [27] in the windowed Fourier transform [28].

4. COMPARISON BETWEEN MEASUREMENT AND SIMULATION

To validate the measurements, rigorously simulated data have been obtained with the RCWA method [2,29]. As input for the simulations, we used the intensity and phase of the field as measured in the plane before the objective (MO in Fig. 2). In Figs. 4 and 5, the experimental and simulated intensities and phases of the scattered far field are shown for three different sets of polarization between the incident and the scattered light, namely $\eta -\eta$, $\eta -\xi$, and $\xi -\xi$. In the case of a nonbirefringent grating, we assume $\eta -\xi$ and $\xi -\eta$ to be identical. Since in our example the scattered far field consists only of the zeroth order, we immediately obtain the complex elements of the scattering matrix.

 

Fig. 4. Measured (left) and simulated (right) intensities of the far field scattered by a grating illuminated by a focused field for different combinations of input and output polarizations. The incident wavelength is 633 nm, and $\mathrm{NA}=0.4$. The grating parameters are given in Table 1.

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Fig. 5. Phase of the scattered far field retrieved from measurements (left) and simulations (right) for different combinations of input and output polarizations. The incident wavelength is 633 nm, and $\mathrm{NA}=0.4$. The grating parameters are given in Table 1.

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The measured far-field intensities and phases show a good match between experiments and simulations. The differences are attributed to extra insertion losses and noise that have not been taken into account in the simulations. In Fig. 5, the deviation between experimental and simulated phase for the $\eta -\xi$ case for normal incidence can be attributed to the low intensity levels in the far fields used to reconstruct the phase map. Finally, we add the ability to obtain the phase maps to define the value of the technique for phase sensitive scatterometry. With phase and amplitude knowledge of the scatterer, all field components (in our case, reflected) from the object within the NA of the optical system, except a constant phase, are obtained.

5. CONCLUSIONS

In this paper, we demonstrate a robust, fast, and reliable way to obtain the entire scattering matrix of a periodic object by measuring the phase of the scattered field from a periodic object at all angles within the numerical aperture of the system using ICFS. With ICFS, one is able to extend the present method of optical scatterometry to its maximum potential. The method presented here provides information on the scattering matrix resolved over polarization for all scattering angles, which can be extended to higher diffraction orders if they exist. Although the results presented above are for small NA (0.4) and long wavelength (633 nm), they can be scaled to higher NA and shorter wavelengths. This approach is not limited to periodic objects and thus can be applied when scattered light from an arbitrary scatterer is used to retrieve information about it. Because the complex scattering matrix provides all possible information, we believe that this method can also be used to set the limits of optical scatterometry in different applications such as object parameter retrieval, detection of (sub-wavelength) particle contamination, defect detection, and surface characterization. Furthermore, depending on the features that are to be retrieved, one can determine the measured data that contain the most sensitive part of the information for this feature and discard the other data. In this way, the speed of scatterometry can be increased.