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

Structured-illumination microscopy allows widefield 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 paper, we analyse the noise properties and achievable resolution of linear and nonlinear 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 photoswitching of fluorescent labels. We compare the performance of two alternative implementations, using either two-dimensional (2D) illumination patterns or sequentially rotated one-dimensional (ID) patterns. We show that 1D patterns are advantageous in the linear case, and that in the nonlinear case 2D patterns provide a slight signal-to-noise advantage under idealised conditions, but perform worse than 1D patterns in the presence of nonswitchable fluorescent background. LAY DESCRIPTION: Structured-illumination microscopy (SIM) is a high-resolution light microscopy technique that allows imaging of fluorescence at a resolution about twice the classical diffraction limit. There are various ways that the illumination can be structured, but it is not obvious how the choice of illumination pattern affects the final image quality, especially in view of the noise. We present a detailed performance analysis considering two illumination techniques: sequential illumination with line-gratings that are shifted and rotated during image acquisition and two-dimensional (2D) illumination structures requiring only shift operations. Our analysis is based on analytical theory, supported by simulations of images considering noise. We also extend our analysis to a nonlinear variant of SIM, with which enhanced resolution can be achieved, limited only by noise. This includes nonlinear SIM based on the light-induced switching of the fluorescent molecules between a bright and a dark state. We find sequential illumination with line-gratings to be advantageous in ordinary (linear) SIM, whereas 2D patterns provides a slight signal-to-noise advantage under idealised conditions in nonlinear SIM if there is no nonswitching background.

Interferometer-based structured-illumination microscopy utilizing complementary phase relationship through constructive and destructive image detection by two cameras.

In an interferometer-based fluorescence microscope, a beam splitter is often used to combine two emission wavefronts interferometrically. There are two perpendicular paths along which the interference fringes can propagate and normally only one is used for imaging. However, the other path also contains useful information. Here we introduced a second camera to our interferometer-based three-dimensional structured-illumination microscope (I(5)S) to capture the fringes along the normally unused path, which are out of phase by π relative to the fringes along the other path. Based on this complementary phase relationship and the well-defined phase interrelationships among the I(5)S data components, we can deduce and then computationally eliminate the path length errors within the interferometer loop using the simultaneously recorded fringes along the two imaging paths. This self-correction capability can greatly relax the requirement for eliminating the path length differences before and maintaining that status during each imaging session, which are practically challenging tasks. Experimental data is shown to support the theory.

Phase-retrieved pupil functions in wide-field fluorescence microscopy.

Pupil functions are compact and modifiable descriptions of the three-dimensional (3D) imaging properties of wide-field optical systems. The pupil function of a microscope can be computationally estimated from the measured point spread function (PSF) using phase retrieval algorithms. The compaction of a 3D PSF into a 2D pupil function suppresses artefacts and measurement noise without resorting to rotational averaging. We show here that such 'phase-retrieved' pupil functions can reproduce features in the optical path, both near the sample and in the microscope. Unlike the PSF, the pupil function can be easily modified to include known aberrations, such as those induced by index-mismatched mounting media, simply by multiplying the pupil function by a calculated aberration function. PSFs calculated from such a modified pupil function closely match the corresponding measured PSFs collected under the aberrated imaging conditions. When used for image deconvolution of simulated objects, these phase-retrieved, calculated PSFs perform similarly to directly measured PSFs.