A modal method is concerned with the modes excited, propagating, and coupled out into diffraction orders of a grating.
Compared with the well-known rigorous-coupled-wave algorithm (RCWA), the modal method is less recognized. While
the RCWA is a pure-numerical method, the modal method reveals a clear physical picture of the modes inside the grating.
When a grating has a large period, it usually has too-many modes that are excited and propagating, and its analysis
would be too complex. When a grating has a subwavelength period or a close-to-wavelength period, a few modes will be
excited and propagating. When a few modes are concerned in the analysis, we have a simplified modal method.
We have developed a simplified modal method to explain diffraction due to a deep-etched fused silica grating. Here
the “deep-etched” means that the grating has a high ratio of deep-etched depth to the groove width. When a deep-etched
fused silica grating has a subwavelength period, there are a few (one to two) lower modes that are propagating modes,
and higher-order modes are evanescent. We have developed an average effective-index concept to describe a triangular
deep-etched grating, and obtained simple analytical equations to describe a three-port beam-splitting grating. These
analytical equations are impossible to obtain with the pure-numerical RCWA. This simplified modal method should be a
useful tool for designing a variety of subwavelength deep-etched fused silica gratings for practical applications.