Resolving power of a microscope? I was reading up on the resolution of a microscope. I read (in some lecture notes) that the size of the limiting spot size is $1.22 \lambda f/W$. But that the smallest resolvable feature has a size $\lambda f/W$. Why are these different? Should they not be the since one is simply the reverse of the other? 
This appears around slide 99 in this document.
 A: Don't get too precious over the term "resolution". There are many ways to define it, and indeed ultimately what you resolve with a microscope gets down to what measurement signal to noise ratio you can achieve. With a perfectly clean signal, you can deconvolve the lens's point spread function from your image and resolve features smaller than what the simple formulas imply. The "diffraction limit" is not a hard limit since it is a lowpass spatial filtering: you can reverse the lowpass by deconvolution if the noise levels allows. Practically, though, you can seldom do this. Often when you work out the number of photons per second coming from each resolvable volume in microscopy, it's surprisingly low and hence the quantum limit is going to hit you. 
The first formula is found by measuring the diameter of the first zero in the perfect, unapodised point spread function ("Airy Disk") given by $\frac{J_1(k\,\eta\,r)}{k\,\eta\,r}$, where $\eta$ is the numerical aperture. The Bessel function $J_1$ has its first zero at 3.83, hence the "resolution" measured this way is $2\times 3.83/(\eta\,k)$, which yields your first formula if you multiply it out (since $2\times 3.83 / 2\pi = 1.22$). 
Other ways of defining resolution are by quoting the point spread function's full width half maximum. Still others quote  diameter that encircles some fraction of the power through the diffraction limited spot, and this fraction can vary: $95\%$, $1-e^{-1}$ and $1-e^{-2}$ are all common.
A: There is no contradiction: Your first answer, with the numerical factor 1.22, is a measure of the width of the diffraction spot from a circular aperture (e.g. from a round lense). Historically, the smallest resolvable feature has usually been defined not as this size, but as the closest distance between point sources that create an image that still has a local minimum between them. If you take this to the extreme, you will arrive at the Abbe limit (minimum distance of $\lambda/2$ for point sources to still appear as two points without post-processing).
Note that for a microscope, unlike the telescope discussed in the particular slide you are referring to, these limits are routinely circumvented at least slightly. There are at least three approaches:


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*Immersion objectives. By adding an immersion oil between your object and the objective lense, you increase the optical density $n$ and hence decrease the wavelength according to $\lambda = \lambda_\mathrm{vacuum} / n$, which is a relatively easy way to get some improvement and very commonly used even in cheap microscopes.

*Multi-photon excitation microscopy. Rather than directly illuminating, you use a fluorescent dye and illuminate that with multiple lasers that must all overlap to create fluorescence. Whilst they are of longer wavelength, the overlap requirement (or a nonlinear intensity dependency) can be used to improve resolution.

*Near-field microscopy. If you forgo your lense optics entirely and either illuminate or observe through a waveguide (i.e. a fiber) brought very close to you object, you can "focus" essentially as tightly as the hole you make in your waveguide for light to get in or out, at least if you get close enough (much closer than the wavelength to gain in resolution).
