Signal, noise, and resolution in linear and nonlinear structured-illumination microscopy

Structured-illumination microscopy allows wide field fluorescence imaging with resolution beyond the classical diffraction limit. Its linear form extends resolution by a factor of two, and its nonlinear form by an in-principle infinite factor, the effective resolution in practice being determined by noise. In this article, we analyze the noise properties and achievable resolution of linear and non-linear 1D and 2D patterned SIM from a frequency-space perspective. We develop an analytical theory for a general case of linear or nonlinear fluorescent imaging, and verify the analytical calculations with numerical simulation for a special case where nonlinearity is produced by photo switching of fluorescent labels. We compare the performance of two alternative implementations, using either two-dimensional illumination patterns or sequentially rotated one-dimensional patterns. We show that one-dimensional patterns are advantageous in the linear case, and that in the nonlinear case two-dimensional patterns provide a slight signal-to-noise advantage under idealized conditions, but perform worse than one-dimensional patterns in the presence of non-switchable fluorescent background.