A general theory of far-field optical microscopy image formation and resolution limit using double-sided Feynman diagrams

Naoki Fukutake

doi:10.1038/s41598-020-73584-1

Abstract

Optical resolution of far-field optical microscopy is limited by the diffraction of light, while diverse light-matter interactions are used to push the limit. The image resolution limit depends on the type of optical microscopy; however, the current theoretical frameworks provide oversimplified pictures of image formation and resolution that only address individual types of microscopy and light-matter interactions. To compare the fundamental optical resolutions of all types of microscopy and to codify a unified image-formation theory, a new method that describes the influence of light-matter interactions on the resolution limit is required. Here, we develop an intuitive technique using double-sided Feynman diagrams that depict light-matter interactions to provide a bird’s-eye view of microscopy classification. This diagrammatic methodology also allows for the optical resolution calculation of all types of microscopy. We show a guidepost for understanding the potential resolution and limitation of all optical microscopy. This principle opens the door to study unexplored theoretical questions and lead to new applications.

Highlights

Optical resolution of far-field optical microscopy is limited by the diffraction of light, while diverse light-matter interactions are used to push the limit
To generalize Abbe’s image formation theory into a unified theory that can address all light-matter interactions, we expand some concepts as follows: (i) a target to be observed has i-th order linear/nonlinear susceptibility χ(i), (ii) diffraction is replaced by QPM, and (iii) the vacuum field is included as one of the excitation fields
If multiple scattering and depletion occur in non-transparent samples, the acquired image will be blurred to some extent, causing the decrease in signal-to-noise ratio

Summary

Introduction

Optical resolution of far-field optical microscopy is limited by the diffraction of light, while diverse light-matter interactions are used to push the limit. Ernst Abbe established the image formation theory in optical microscopy and derived the well-known optical resolution formula, d = λ/2NA This resolution limit corresponds to a frequency cut-off of 2NA/λ, where λ is the wavelength of light and NA is the numerical aperture of the microscope objective[1]. When nonlinear susceptibility χ(i) are included, the resolution limit may surpass 2NA/λ and the missing cone problem can be overcome, where i≧2 for higher-order light-matter interactions[16] This implies that the higher-order interactions, even linear fluorescence, which is an χ(3)-derived interaction, cannot be addressed using Abbe’s formula. We formulate a new universally applicable theory, and derive the rules for calculating the resolution limit for all optical microscopy techniques that use arbitrary light-matter interactions. To achieve the universally applicable theory, we reframe the diffraction as the lowest order quasi-phase-matching (QPM)[21], and include the vacuum field as one of the excitation fields responsible for the light-matter interaction

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Results

Conclusion