Let me start with a detour into epifluorescence imaging and even sample illumination, before coming back to the main questions. Bear with me please.
For epifluorescence imaging, one desires even illumination of the sample. I picture this as collimated light exiting the front of the objective. To achieve this, as I understand it, one should form an image of the light source (used to be a filament) into the back focal plane (BFP) of the objective. When this is accomplished, one expects a collimated beam out the front and even illumination of the sample. Adding a diffuser somewhere between light source and objective might help even things out too.
Q1: Am I correct in thinking that for even illumination at the sample, one should see an image of the light source at the BFP?
On a typical epi-fluorescence microscope one used to have a halogen lamp or some other light source (maybe coupled via a liquid light guide). These days we're moving away from halogen lamps, and many companies (CoolLED, Thorlabs, Prizmatix), including microscope manufacturers (e.g. Zeiss Colibri), offer LED light sources intended to be attached to the epi port of a microscope, in place of a typical halogen lamp housing. I'd like to focus on the case of an LED, but let's start with the more common lamp housing (e.g. halogen).
Most microscopes have an "epi illumination port", designed originally to attach a lamp housing. Those lamp housing often had fine controls to adjust the position of the light source laterally, and I think sometimes axially also. The microscope user is supposed to periodically check that the light source (filament) is centered and focused into the BFP (by removing the objective and e.g. holding a piece of paper in its place); the adjustment screws on the lamp housing allow this to be fine-tuned. My mental image of this optical configuration is "inverse Koehler", for lack of a better term. Normally I think of a point source at the sample emitting rays that enter the front of the objective (at maximum angles determined by the NA), which then exit the back of the (infinity-corrected) objective as collimated light. Yet for epi illumination, (I believe) the "point" source should be focused into the center of the BFP, and collimated light exits the front.
The LED light sources marketed for microscope epi illumination in place of traditional halogen lamps typically include optical components to generate a (roughly) collimated beam. But is that what we want? When I put a collimated LED onto the epi port, then remove the objective and observe the approximate BFP location, I typically do not see an image of the LED emitter (as I would see the filament image for a properly aligned halogen lamp). Depending on the microscope and LED model, sometimes it is possible to obtain a focused image of the LED emitter at the BFP by adjusting the axial position of the LED - moving it back further away from the port (this requires custom hardware). Alternatively, one might be able to adjust the axial position of the collimating lens of the LED module to achieve the same goal - for some models, this works, but it's quite tedious. A few models (e.g. CoolLED pE-300) have a mechanism for adjusting the collimation, a simple sliding lever: in that case, the power delivered to the sample varies quite significantly as one adjust the slider. I have not personally checked whether that LED model lets the source (aka LED emitter) be focused at the BFP, for some slider position.
Q2: So my second question, really, is: what's the deal with all these "epi port LED sources" for microscopes? I don't think the original halogen lamp housings were producing a collimated beam at the port, so why do they make the LEDs that way? Is there any specific reason to believe that microscopes "expect" a collimated input beam at the epi port? Another way to pose the question: does mounting a collimated LED at the epi port of a microscope result in even illumination at the sample?
On a typical epi-fluorescence microscope, however, one does not use a laser as the light source, but ...
Most spot-size calculators and discussions assume a Gaussian beam entering the back focal plane of the objective. The relevant formula is: Wd = 2*Wr = 4 * lambda / pi * F / D Wd/r = waist diameter or radius; F = focal length; D = beam or aperture diameter
This seems appropriate if one sends e.g. a laser beam through a microscope and into the back of the objective. I assume that if the beam is larger than the back aperture, the relevant value D is the back aperture diameter, and if the beam is smaller than the back aperture, one should use the 1/e2 beam diameter for D.
Some related references: http://www.calctool.org/CALC/phys/optics/f_NA (note, seems to have a bug; Rayleigh length is 4x too large) How to calculate spot size of a laser focused through a microscope objective Focal length vs working distance in an infinity corrected objective Best collimation at focal point of lens? https://www.olympus-lifescience.com/en/microscope-resource/primer/techniques/fluorescence/anatomy/fluoromicroanatomy/ (good diagram of typical epi scope, and of the epi illuminator arm in Fig. 4)