Our new theory covers both the Kohler illumination systems and laser excitation systems. Until now, the image formation theories do not exist that attempt to generalize the Abbe’s formula into a more unified theory of light-matter interactions, except for our previous works 16, 17, 18, which address only laser microscopy. Thus, we need to expand the diffraction theory in χ (1)-derived interactions to also the high-order interactions. This frequency cutoff happens to become equal to that of bright-field microscopy, which does originate from the diffraction. A typical example of inappropriate interpretation is that the frequency cutoff of 2NA/λ of wide-field fluorescence microscopy originates from the diffraction. Nevertheless, we encounter the implication of incorrectly applying Abbe’s formula to χ ( i)-derived interactions ( i ≧2). However, we can expand the concept of diffraction to even fluorescence by including the vacuum field in the light-matter interaction. While the Abbe’s formula is based on the diffraction of light, the fluorescence microscopy theory does not apply the concept of diffraction because fluorescence appears to be unrelated to the diffraction. The image formation theory of fluorescence microscopy exists, but it is formulated by using different approach. Indeed, fluorescence confocal microscopy possesses a frequency cutoff of 4NA/λ 10, presuming that the wavelengths of the excitation beam and fluorescence are equal. This implies that the higher-order interactions, even linear fluorescence, which is an χ (3)-derived interaction, cannot be addressed using Abbe’s formula. 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. Recently, many types of optical microscopy have been developed that use a variety of light-matter interactions, such as two-photon excited fluorescence 11, second-order harmonic generation 12, third-order harmonic generation 13, coherent anti-Stokes Raman scattering 14, stimulated Raman scattering 15, and so on.
Although Abbe’s definition of resolution limit is still used as the standard, it is not well known that the frequency limit 2NA/λ applies only to microscopy techniques that utilize χ (1)-derived interactions. The missing cone expresses the frequency region that cannot be resolved, which causes poor optical resolution in the z (optical-axis) direction. Considering three-dimensional (3-D) optical resolution in transmission microscopy that uses χ (1)-derived interactions, it is known that the missing cone problem exists in the spatial-frequency domain 10. These microscopy use only χ (1)-derived light-matter interactions, such as linear absorption, transmission, and reflection, where χ (1) is the electrical susceptibility. Abbe’s theorem is obeyed by classical microscopy techniques such as bright-field microscopy 2, phase contrast microscopy 3, 4, differential interference microscopy 5 and dark field microscopy 6, as well as by newer microscopy techniques such as relief contrast microscopy 7, digital holographic microscopy 8, and optical coherence tomography (OCT) 9. 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. This principle opens the door to study unexplored theoretical questions and lead to new applications.Įrnst Abbe established the image formation theory in optical microscopy and derived the well-known optical resolution formula, d = λ/2NA.
We show a guidepost for understanding the potential resolution and limitation of all optical microscopy. This diagrammatic methodology also allows for the optical resolution calculation of all types of microscopy. 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. 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. 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. 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.
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