The size of microscopic particles can be determined using laser light, which is a kind of electromagnetic wave. The roughness of surfaces can affect the result of size measurements of other objects and is also a limitation in industrial applications such as the manufacture of optical instruments. Light is directed towards the microscopic particles or rough surfaces and the resulting distribution of scattered light is then measured. These measurements contain the information needed to determine the size of the particles or the characteristics of the rough surfaces. The distribution of the light scattered by typical particles and surfaces is calculated, and this process can be inverted to determine the properties of the scatterer from the set of measurements.

Electromagnetic scattering is governed by the partial differential equations known as Maxwell's equations. These equations can be solved numerically either in their original form or after they are converted to integral equations. When the wavelength of the laser light used in the measurements is about as large as the diameter of the particles or the root-mean-square roughness of the surface, the exact equations have to be used. If the wavelength is much larger than the dimensions of the particles or surface roughness, approximate theories can be used to simplify the calculations. Both of these situations occur in measurements carried out in this division.

imple numerical integration can be used to calculate the light distributions when approximations are valid. Otherwise, the numerical solution of the Maxwell equations in integral form is implemented by converting these equations into linear algebraic equations. Integral equations have some advantages over differential equations, mainly consisting of the incorporation of the radiation condition and the flexibility in the choice of domains of integration. For homogeneous scatterers, the unknowns for integral equations are located on the interfaces and do not need to be distributed throughout the volume, greatly reducing their number. Two-dimensional problems such as monochromatic plane waves incident on infinite strips on a substrate can be reduced to the solution of scalar Helmholtz equations. Full three-dimensional problems are more complicated and calculations are limited by the memory available on the computers used to solve these problems.

Compute the distribution of light scattered by pigments in rough coatings. Compute the distribution of light scattered by parallel strips on a substrate. Compare power spectral densities of surface roughness determined using a variety of profiling instruments, as well as a light scattering instrument.