Yes and my problem is why do people define the point spread function as such without considering for these effects? "...but it could depend on how far the observer is from the source" I believe this is temporal coherence of the individual emitters. Which further adds to the spatial incoherence and hence my confusion

"Spatially coherent = all photons same direction" Why is that? Think of spherical wave. The k vectors of all points are converging/diverging. Spatial coherence need not be only for plane wave.

Coherent (spatial) source is monochrome doesn't mean monochrome source are spatially coherent. If I place a narrow band filter in front of a light bulb the light is not spatially coherent right? That said, I don't believe we can consider incoherent source as superposition of many different wavelengths while each of them having spatial coherent and the result is just a summation. Is this wrong?

Since when did I disregard point source as divergent? I just can't accept why they are treated as divergent and collimated AT THE SAME TIME. I don't really know what you are getting at for the rest of the statement. It seems you are saying the left and right setup don't happen at the same time. If that is the case, you will be wrong. In all the articles I read they occur simultaneously hence my confusion. "And no, lenses do make fourier transforms, no need for coherence" my question is why doesn't it need coherence but your response is "yeah it doesn't" ...

Ok let's start with your drawing. I don't really know what you are trying to achieve with this. Clearly LED1 forms image at plane 1 while LED2 at plane 2. So what is this suppose to explain? Divergence affects focal? Yes I know that and in fact that is my question from the start. Plane 1 is like the field stop of the system so at plane 2 is the light focused or collimated? Answer depends on the input but I only have 1 source. You see the question?

Are you suggesting that there are collimated and divergent rays coming out of a light source?

I see that are some edits but I still fail to see how this address the question. You are saying more light reaches the sample via the condenser lens which ends up as a background in the sensor. So what about the imaging part that the field stop is imaged on the sensor? In that path the light immediately after the aperture stop is collimated. So what is it? collimated or divergent? Yeah I know its both. The question is why is it both. Imaging in geometric optics always treat object as point sources. If it is placed at infinity it is collimated but never both.

Coming back to your comment, I don't see how you are answering the question. Basically you are saying Kohler is not critical illumination. "The light can be collimated or divergent, or convergent" This is my problem. The illumination and imaging path exist SIMULTANEOUSLY. Its not that they are separate extreme cases catering to light source of different divergence. "... the features in your sample are new point source emitters" agree on this but that is not my question. I am ok with this. My question is really why the light source is drawn as divergent in 1 case and collimated in the other

This is another thing I heard almost everywhere but don't understand why "lens does Fourier transform". Where does that come from? Fourier optics? In Fourier optics the first thing textbooks say is plane wave i.e. coherent light source and if you really think about it, it has to be coherent light else you can't sum the light to approximate the Fourier expression. That being the case, why is this relevant? The light source here need not be coherent and most of the time it isn't.

This really is another way of saying that are always 2 paths although my question is really why there are 2. Let's look at the light source, why is it divergent/parallel like in your answer? What does that mean? Are you saying if I have a point source it can be both divergent and collimated? Or is this some special treatment for extended source? Shouldn't extended source be just considered as multiple point sources? Also, what does it mean when you say "diverging/parallel beam of light"? If I have a single lens, I will have 2 focal points? 1 for divergent, 1 for parallel??

I see. I guess you are right, this makes sense

Usually only 1 PSF is used in image formation theory for widefield microscopy since the illumination is constant. However in confocal microscope, the illumination is localized so it is represented by a PSF (which is what almost every single text I read talks about: resolution is not infinite because of diffraction limit). If it helps, we can use the idea used in Image Scanning Microscopy (PRL 104, 198101 (2010)) to think about the PSF. Illumination PSF describes the probability portion of the sample is illuminated while detection PSF is detection probability by detector from a point source.

I assumed it was noise but I thought that less than 2 times improvement in resolution is so much off from infinite. I am not sure about the second point. Why is the inverse operation not unique? Do you mean because some points are 0 in the k-space? Or do you mean we can't find unique solution?

Image formation is always explained using I(r)=PSF(r)*O(r) where PSF(r) is the point spread function, O(r) is the object emission and * is the convolution operator. I have never seen any other equations. Are there other models?

Anyway, how can I understand this problem using Fourier transformation? That the PSF can be explained with Fourier transform? Won't that be my first point? As in this explain the illumination spot size but not the detection? Regarding the second question, why is convolution more 'natural'? It seems cross correlation is more natural. I don't understand why you have to flip the PSF/object.