Nanostructures—objects that are a few tens of nanometers in size—are common in the semiconductor industry. Nowadays they can be easily fabricated using lithography and nanoimprinting techniques. Memory modules that use 32nm circuit patterns are already produced for commercial products. In addition, nanoscale structures such as wire-grid polarizers, antireflection structures, and photonic crystals can be used as optical elements in semiconductors. To produce these structures, it is necessary to accurately measure their surface profiles. Scanning electron microscopy and atomic force microscopy are suitable methods for making these measurements at the nanometer scale. However, the time involved can be very slow for areas of several hundred square micrometers, thus making these techniques impractical.

Scatterometry is a powerful, rapid, and non-contact method for measuring the surface profiles of nanostructures.^{1–5} Light at a variety of wavelengths is used to illuminate a target, and the level of the returned signal after interaction is measured. Experimentally measured values are compared with those calculated using optical parameters. In cases where the period of the structure's profile is longer than the beam's wavelength, no diffraction pattern is observed (see Figure 1). Instead, information about the surface is presented as a structured birefringence (splitting of light into two rays as it passes through a material), where the beam is observed as the zeroth order. The light's polarization state depends on the surface profile of the sample, but it is not easy to determine this uniquely. A proposed scatterometry method that is based on ellipsometry (a range of techniques used to characterize material properties based on optical features) evaluates a surface profile from the phase difference and the complex reflection amplitude of the reflected beam.^{1}

**Figure 1. **Effective scatterometry methods for different structure periods (d) and wavelengths (λ).

We use a scatterometry technique that is based on the Mueller matrix (a calculus method for manipulating vectors that represent the polarization of light) to evaluate the surface profiles of nanostructures.^{2–5} Our method can effectively assess the depolarization of light caused by scattering defects, in addition to fully polarized properties such as structured birefringence, optical rotation, and diattenuation.

We have compared the surface profile results obtained by applying rigorous coupled-wave analysis (RCWA)^{6} and the boundary element method to a nanostructure sample, with the results from our Mueller matrix method. Based on these results we designed a line-type spectroscopic Mueller matrix polarimeter that can be used to obtain specific polarization properties.^{7, 8}

The experimental setup of our polarimeter is shown in Figure 2. Collimated (parallel) white light from a halogen lamp, connected by optical fibers, passes through a sample placed on a dual rotating quarter-wave plate between a crossed polarizer and an analyzer. The intensity of the transmitted light is measured by a spectral line imaging camera system that consists of a CCD camera, grating, and prism. The camera operates over a wavelength range of 300–900nm. The incident and detection arms of the experimental setup can easily be changed in the 30–90° range.

**Figure 2. **The Mueller matrix (m) spectroscopic method and experimental setup. FTDT: Finite-difference time-domain. QWP: Quarter-wave plate. σ: Sample variance. k: Sampling number wavelength. m_{ij − exp}: Experimental results of the Mueller matrix. m_{ij − the}: Simulated results of the Mueller matrix.

We used our experimental setup and calculation technique to measure the surface profile of a wire-grid polarizer that had a period of 155nm and a height (H) of 200nm (see Figure 3). We used RCWA to analyze 25 different cases where H = 10, 100, 200, 300, or 400nm; L (width) = 31, 62, 77.5, 93, or 124nm; and for wavelengths of 480 or 680nm. We compared the sample variance of 16 elements of the Mueller matrix for the measured and calculated results. The minimum sample variance (marked by circles in Figure 3) indicated three possibilities for the dimensions of the wire-grid polarizer: H = 400nm and L = 62nm; H = 100nm and L = 62nm; or H = 200nm and L = 77.5nm. Our calculated and measured values were therefore in agreement.

**Figure 3. **Surface profile measurement results for a wire-grid polarizer. The dotted circles denote regions of minimum sample variance. H: Height. L: Width. SV: Sample variance.

We have successfully demonstrated the effectiveness of using a Mueller matrix technique to measure surface profiles of nanostructures. We intend to use this method to measure periodic structures and to identify defects in such constructs. This development will require an additional analysis step that considers depolarization of light due to scattering.

Yukitoshi Otani

Center for Optical Research and Education (CORE)

Utsunomiya University

Utsunomiya, Japan

Yukitoshi Otani is a professor at CORE. Prior to this position, he was an associate professor at the Tokyo University of Agriculture and Technology and a visiting professor at the University of Arizona. He has been a Fellow of SPIE since 2010.

Yasuhiro Mizutani

Institute of Technology and Science

University of Tokushima

Tokushima, Japan

References:

1. B. K. Minhas, S. A. Coulombe, S. S. H. Naqvi, J. R. McNeil, Ellipsometric scatterometry for the metrology of sub-0.1-μm-linewidth structures, *Appl. Opt.* 37, p. 5112-5115, 1998.

2. T. Novikova, A. D. Martino, P. Bulkin, Q. Nguyen, B. Drevillon, Metrology of replicated diffractive optics with Mueller polarimetry in conical diffraction, *Opt. Express* 15, p. 2033-2046, 2007.

3. Y. Otani, T. Kuwagaito, Y. Mizutani, N. Umeda, Nano-structure measurement by Mueller matrix polarimeter and RCWA, *2007 SEM Annu. Conf. Exposition Exp. Appl. Mech.*, p. 1-4, 2007.

4. Y. Otani, T. Kuwagaito, Y. Mizutani, Surface profile detection with nanostructures using a Mueller matrix polarimeter,

*Proc. SPIE* 7063, p. 70630Y, 2008.

doi:10.1117/12.798143
5. Y. Mizutani, Y. Otani, Ellipsometry at the nanostructure, *Ellipsometry at the Nanoscale*, p. 313-323, Springer, 2013.

6. M. G. Moharam, T. K. Gaylord, Diffraction analysis of dielectric surface-relief grating, *J. Opt. Soc. Am.* 72, p. 1385-1392, 1982.

7. R. M. A. Azzam, Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal, *Opt. Lett.* 2, p. 148-150, 1978.

8. S.-Y. Lu, R. A. Chipman, Interpretation of Mueller matrices based on polar decomposition, *J. Opt. Soc. Am. A* 13, p. 1106-1113, 1996.