Light scattering is widely used by scanning surface inspection tools to inspect materials, such as silicon wafers, flat panel display substrates, data storage media, and optics, for defects, particles, and surface roughness. These instruments often direct collimated or lightly-focused light at a sample and collect scattered light with one or more large optics. The optics collect light over significant solid angles in order to maximize sensitivity. When designing or calibrating these instruments, it helps to be able to predict the signal for different geometric conditions and model parameters.
The BRDF characterizes the directional dependence of the scattering by a material. The BRDF is given by \begin{equation} {f_{_{\rm{r}}}}({\theta _{\rm{i}}},{\theta _{\rm{s}}},{\phi _{\rm{s}}}) = \mathop {\lim }\limits_{\Omega \to 0} \frac{{{\Phi _{\rm{s}}}}}{{{\Phi _{\rm{i}}}\Omega \cos {\theta _{\rm{s}}}}} \end{equation} where ${\Phi _{\rm{s}}}$ is the power scattered into solid angle $\Omega$, centered on polar angle $\theta_{\rm{s}}$ and azimuth angle $\phi_{\rm{s}}$, and $\Phi_{\rm{i}}$ is the power incident on the sample at an angle $\theta_{\rm{i}}$. The angles $\theta_{\rm{i}}$, $\theta_{\rm{s}}$, and $\phi_{\rm{s}}$ are defined in Fig. 1. The function $f_{\rm r}(\theta_{\rm i},\theta_{\rm s},\phi_{\rm s})$ also depends upon the polarization state of the incident light and polarization sensitivity of the detection system. Any optic collecting light over a finite solid angle measures a reflectance given by \begin{equation} \rho (\Omega ) = \int\limits_\Omega {{f_{\rm{r}}}({\theta _{\rm{i}}},{\theta _{\rm{s}}},{\phi _{\rm{s}}})\cos {\theta _{\rm{s}}}\,\sin {\theta _{\rm{s}}}\;{\rm{d}}{\theta _{\rm{s}}}\;{\rm{d}}{\phi _{\rm{s}}}} \end{equation}
The MIST program is designed to perform this integration using any of the BRDF models provided in the SCATMECH library and for geometries specified by the user. Furthermore, MIST allows the user to vary parameters in the model or in the definition of the solid angle, providing the user with the dependences on those parameters.
Besides using it as a design tool, one application of MIST is the development of calibration curves for an instrument. For example, the absolute response of an instrument to spherical particles can be accurately determined as a function of the particle size using the SCATMECH model Bobbert_Vlieger_BRDF_Model. If a set of reference particles are used to calibrate the instrument, then these calibration curves can be used to interpolate (or even extrapolate) to other particle sizes. The advantage of using a model-based calibration curve, over using non-physically-based interpolation techniques, is that it is more accurate and the calibration should not change if the set of reference particles changes. Furthermore, a smaller number of reference particles are needed for calibration, and over sampling of the calibration curve provides information about uncertainties in the measurement and can compensate for uncorrelated uncertainties in the reference particle diameters.
FIGURE 1 The scattering geometry and angle convention used by the calculations. The incident angle is $\theta_{\rm i}$. The scattering direction is defined by polar angle $\theta_{\rm s}$ and azimuth angle $\phi_{\rm s}$.
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Latest MIST Version: 4.10 (October 2017)