Method and system for measuring differential scattering of light off of sample surfaces

A computerized method for estimating scattering of electromagnetic radiation from a surface, the method comprising:

• providing a distribution expression that includes a first integral over source solid angle, a second integral over a sample area, a third integral over detector solid angle, and an integrand that includes a differential-scattering profile;
• approximating the first and second integrals to be the second integral, wherein source electromagnetic radiation is approximated to be collimated,
• approximating third integral to be one based on detecting the electromagnetic radiation scattered from the surface at an imaging detector;
• transforming the coordinates of the second integral over the sample area to first and second dimensions in cosine space to form a fourth integral, wherein the surface is approximated to be shift invariant;
• integrating over the first dimension of the fourth integral;
• differentiating the fourth integral with respect to the second dimension to generate the differential-scattering profile;
• and
• generating an optical system design based on the differential-scatting profile.

An optical system comprising:

• a collimated beam of electromagnetic radiation configured to illuminate a sample surface, the sample surface being shift invariant;
• an imaging detector configured to collect electromagnetic radiation scattered from the sample surface, the imaging detector configured to collect the scattered electromagnetic radiation at a plurality of scattering angels to generate a scattering profile;
• and
• a computer device configured to generate an estimated differential-scattering profile and compare the scattering profile and the estimated-differential-scattering profile to generate an optical system design, wherein the estimated-differential-scattering profile is a continuous solution of an differential model of spectral scattering and non-spectral scattering derived from a deconvolution of a bidirectional reflectance distribution function (BRDF),
• wherein:
• an expression for the BRDF includes a first integral over a source solid angle, a second integral over the sample surface, a third integral over detector solid angle, and an integrand that includes the estimated-differential-scattering profile;
• the first and second integrals are approximated to be the second integral based on the source electromagnetic radiation being in the form of the collimated beam;
• the third integral is approximated to be one based on the detector being an imaging detector;
• the second integral is transformed from an integral over detector solid angle to a fourth integral over first and second dimensions in cosine space based on the
• sample surface being shift invariant;
• and
• the fourth integral is integrated with respect to the first dimension and deconvolved with respect to the second dimension to generate the estimated differential-scattering profile.