How is electromagnetic scattering used in metrology?
The products of the semiconductor industry contain features such as lines and trenches that are gradually being reduced in size, so that now their dimensions are often comparable to the wavelength of visible light. Optical microscopes have advantages in measuring these features, but simulation of the image is required to obtain an accurate result. Also the size of microscopic particles can be determined using laser light. The roughness of surfaces can affect the result of size measurements and is also a limitation in industrial applications such as the manufacture of optical instruments. When light is directed towards the microscopic particles or rough surfaces and the resulting distribution of scattered light is then measured, these measurements might contain enough information to determine the size of the particles or the characteristics of the rough surfaces.
What theory describes the scattering of light?
Electromagnetic scattering is governed by 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 light used in the measurements is about as large as the linewidth, the diameter of a particle, or the root-mean-square roughness of a surface, the exact equations have to be used. If the wavelength is much larger or much smaller than the dimensions of the semiconductor features, particles, or surface roughness, approximate theories can be used to simplify the calculations.
How are calculations carried out?
Simple 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 the incorporation of the radiation condition in special solutions, the Green functions, 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 two coupled scalar Helmholtz equations. Full three-dimensional problems are more complicated and calculations are limited by the speed and memory available on computers.
Mainly computation of optical microscope images of parallel lines on a substrate, trenches in a substrate and spaces in coated substrates, related to overlay and photomask applications in the semiconductor industry. Images of three-dimensional objects such as finite lines are being analyzed and considered for numerical solutions.
Figure 1 shows a profile of relative intensity measured by a transmission optical microscope compared to two simulated profiles obtained with different methods, rigorous coupled waveguide analysis (RCWA) and the solution of integral equations equivalent to Maxwell’s equations as described here. The agreement between the simulated profiles is better than that between them and the measured profile, but there is a definite similarity. The peaks at the junction between the measured line image and the background, which has been normalized to 1, could be due to a value of the illumination numerical aperture that is smaller than the one used in simulations.
Physical Measurement Laboratory (PML)