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Considering green light around 500 nm and a NA of 1, the Abbe limit is roughly = = (0.25 μm), which is small compared to most biological cells (1 μm to 100 μm), but large compared to viruses (100 nm), proteins (10 nm) and less complex molecules (1 nm). To increase the resolution, shorter wavelengths can be used such as UV and X-ray microscopes.
Sparrow's resolution limit is nearly equivalent to the theoretical diffraction limit of resolution, the wavelength of light divided by the aperture diameter, and about 20% smaller than the Rayleigh limit. For example, in a 200 mm (eight-inch) telescope, Rayleigh's resolution limit is 0.69 arc seconds, Sparrow's resolution limit is 0.54 arc seconds.
The formal Rayleigh criterion is close to the empirical resolution limit found earlier by the English astronomer W. R. Dawes, who tested human observers on close binary stars of equal brightness. The result, θ = 4.56/ D , with D in inches and θ in arcseconds , is slightly narrower than calculated with the Rayleigh criterion.
Also common in the microscopy literature is a formula for resolution that treats the above-mentioned concerns about contrast differently. [2] The resolution predicted by this formula is proportional to the Rayleigh-based formula, differing by about 20%. For estimating theoretical resolution, it may be adequate.
By 1978, the first theoretical ideas had been developed to break the Abbe limit, which called for using a 4Pi microscope as a confocal laser-scanning fluorescence microscope where the light is focused from all sides to a common focus that is used to scan the object by 'point-by-point' excitation combined with 'point-by-point' detection. [14]
High-resolution black-and-white film is capable of resolving details on the film as small as 3 micrometers or smaller, thus its cutoff frequency is about 150 cycles/millimeter. So, the telescope's optical resolution is about twice that of high-resolution film, and a crisp, sharp picture would result (provided focus is perfect and atmospheric ...
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In Abbe's 1874 paper, titled "A Contribution to the Theory of the Microscope and the nature of Microscopic Vision", [12] Abbe states that the resolution of a microscope is inversely dependent on its aperture, but without proposing a formula for the resolution limit of a microscope.