# Confocal microscope resolution

I am trying to understand the working principle of a confocal luminescence microscope, coming from a background of gaussian optics. 1. I have the impression that the role of the pinhole on the source side is simply to filter the initially multimode beam and to keep only the TEM$_{00}$ mode (or equivalently to decrease the M factor of the beam to 1), so that the beam can be focused down to $$w=\frac{\lambda f'}{\pi w_{L}}M^{2}=\frac{\lambda f'}{\pi w_{L}}.$$ If this is correct, could we use a single mode optical fiber instead of a pinhole ?

2. I can see two ways to estimate the resolution of the illumination.

a. Considering a gaussian beam focused by a lens of diameter D, the minimum waist is obtained when the incident beam is as large as the lens, leading to $$w=\frac{\lambda2f'}{\pi D}=\frac{\lambda}{\pi NA}\simeq0.32\times\frac{\lambda}{NA}.$$ I assume the same expression holds for a microscope objective, using directly the numerical aperture NA.

b. Considering a plane wave arriving on the objective, Airy diffraction gives rise to a angular diffraction of $\theta=1.22\,\lambda/D$, resulting in the focal plane in a spot of size $$\sim1.22\frac{\lambda f}{D}=0.61\times\frac{\lambda}{NA}.$$

I often come across the second expression but I never found the first one in this context. Is there a reason why ?

3. How is the total resolution of the system estimated ? As the lowest resolution between illumination and imaging ?

To estimate the role of the imaging part of the setup, I tend to consider Fermat principle and imagine what would happen if I were to shine light through the imaging path towards the sample. There again, I would focus the beam into a waist $w'$ which corresponds to the resolution the imaging system can reach. If $w=w'$, I guess this value is the total resolution of the system. I often found a expression as $0.4\times\frac{\lambda}{NA}$ for the overall resolution, but couldn't understand really where it was coming from.