In microscopy, the resolution refers to the ability of the microscope to distinguish between two closely spaced objects as separate entities. The second formula you mentioned is likely referring to the Rayleigh criterion, which states that the minimum resolvable separation (d) between two objects in a microscope is inversely proportional to the wavelength (λ) of the light used and directly proportional to a constant (k) and the numerical aperture (NA) of the microscope objective:
d = (k * λ) / NA
Here's an explanation of why longer wavelengths lead to lower resolution:
Rayleigh Criterion: The Rayleigh criterion is based on the principle of diffraction, which limits the ability of a lens system to produce sharp, well-resolved images. According to the criterion, two point sources are considered resolvable if the central maximum of the diffraction pattern of one source coincides with the first minimum of the diffraction pattern of the other source.
Diffraction Limit: The diffraction pattern created by an objective lens spreads out as a result of the wave nature of light. The smaller the wavelength of light, the smaller the angular spread of the diffraction pattern, and hence the better the resolution.
Longer Wavelength, Wider Diffraction Pattern: As the wavelength of light increases, the diffraction pattern becomes wider, leading to a broader central maximum and less distinct minima. This wider pattern makes it more difficult to distinguish fine details or closely spaced objects, resulting in lower resolution.
Numerical Aperture: The numerical aperture (NA) represents the light-gathering ability of the objective lens. Higher numerical apertures allow for a larger range of angles to be captured, resulting in better resolution. However, even with a high numerical aperture, the resolution will be limited by the wavelength of light used.
Therefore, in microscopy, using shorter wavelengths of light (such as ultraviolet or blue light) provides higher resolution than using longer wavelengths (such as red or infrared light), allowing for better differentiation of fine details in the observed sample.