# What's the difference between a back focal plane and pupil plane?

Yesterday in the lab I asked one of my colleagues whether she knew
what the pupil plane of a microscope objective was. Her answer was
no. I then unknowingly proceeded to give her a description which I now
realize was false. I said that the pupil plane is the plane near the
backside of the objective where the light intensity is proportional to
the Fourier transform of the image. Later that evening, I realized
that this explanation was, in fact, wrong. I had described to her what
is the *back focal plane* for an infinity-corrected objective.

Don't worry, I will correct myself when I see her later today. However, this experience does raise one important question that I have never seriously considered in optics: what is the meaningful difference between the pupil plane of an optical system and its back focal plane?

First, I should point out that there is no difference between a *back*
focal plane and one of the two focal planes of an objective; these are
merely semantics that reflect the fact that we think of the sample as
being in front of the objective. The back focal plane is therefore the
focal plane of the objective located on the side opposite the
sample. Now, everyone who has taken a Fourier optics class has
probably learned the following mantra:

**The focal plane of a lens contains the Fourier transform of the
object.**

Of course, this statement is usually what we remember, but in fact it should look more like this:

**The focal plane of a lens contains the Fourier transform of the
object, except for every other detail we ignore in the math.**

While it's true that the light intensity here reflects the sample's two-dimensional Fourier transform, the general case often contains additional phase factors and expressions to account for the pupil size and apodization (see Goodman, section 5.2). Regardless, it is in the back focal plane where the light intensity contains information on the angular spectrum that makes up the object.

In contrast, the pupil plane contains either the image of the objective's aperture stop or the physical stop itself, depending on what sets the limit on the numerical aperture of the objective in the infinity space (again, see Goodman, appendix B). Moreover, the pupil plane is also used as the reference plane for quantifying the objective's aberrations. This is because, for a diffraction-limited system, the wavefront in the (exit) pupil plane will be a spherical wave converging towards the image of a point source and truncated by the aperture stop (Goodman again discusses this in section 6.1). Deviations from the spherical reference wave serve as the means to quantify the aberrations in a system.

From an experimental point of view, the back focal plane is interesting because it is used in Fourier microscopy, for which many setups have been devised to image it (see Figure 1 in this excellent arXiv submission.) What I really am wondering now is how one would go about imaging the pupil plane as a means of exploring an objective's aberrations, and whether this is at all feasible.

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