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*The focal plane of a lens contains the Fourier transform of the
object.*
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Of course, this statement is usually what we remember, but in fact it
should look more like this:
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*The focal plane of a lens contains the Fourier transform of the
object, except for every other detail we ignore in the math.*
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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 [[https://books.google.ch/books?id=ow5xs_Rtt9AC&printsec=frontcover&dq=goodman+fourier+optics&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMI-5vjt5jGxwIVCMUUCh37ogzP#v=onepage&q=goodman%20fourier%20optics&f=false][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 [[https://books.google.ch/books?id=ow5xs_Rtt9AC&printsec=frontcover&dq=goodman+fourier+optics&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMI-5vjt5jGxwIVCMUUCh37ogzP#v=onepage&q=goodman%20fourier%20optics&f=false][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 ([[https://books.google.ch/books?id=ow5xs_Rtt9AC&printsec=frontcover&dq=goodman+fourier+optics&hl=en&sa=X&ved=0CB0Q6AEwAGoVChMI-5vjt5jGxwIVCMUUCh37ogzP#v=onepage&q=goodman%20fourier%20optics&f=false][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 [[http://arxiv.org/abs/1507.04037][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.