Confocal microscope resolution

Question

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.

Answer 1

You can indeed use a fiber core instead of a pinhole for filtering. Usually the confocal microscope records fluorophore emission which is not really a gaussian but more like a spherical wave. (not exactly as it is a dipole emission...). I would advise to handle the filtering as a geometric effect : light from areas out of focus is more filtered by the pinhole than in-focus light. The pinhole does not really give a better resolution, rather a better optical sectioning on the axial/focus direction.

2. Hmmm... The resolution comes from both your NA at excitation and at collection. The final NA resulting from the convolution of the two point spread functions. The second expressionn is the expression of classical wide field microscopy, but you're right in adding an effect from excitation. For your gaussian calculus it may be accurate but remember that you cannot have a waist of the lens size, as there will be some cropping. Usually it's better to model the PSF and its size by Fourier optics.

3. As you mentioned in 2, both excitation and detection are to be considered. The excitation resolution of 0.32 you calculated for excitation seem really low, as there is inverse invariant propagation of light I would expect similar resolution for excitation and detection. When you combine them you can typically divide the resulting resolution by sqrt(2), getting around 0.4lambda/NA