# Beware the paraxial approximation in microscopy

I have had a difficult time resolving what originally seemed to be an inconsistency between a 2004 research article about estimating microscope pupil functions and the body of knowledge concerning the theory of aberrations in optical systems.

In the article by Hanser, et al.^{1}, the authors utilize an imaging model
for calculating the amplitude point-spread function of an optical
system that incorporates defocus as an exponential term inside the
Fourier transform integral:

where \(k_z = \sqrt{\left( 2 \pi n / \lambda \right)^2 - \left( k_x^2 +
k_y^2 \right)}\). (I changed the notation of some variables used in the
text because the authors confusingly use \(k\) to represent spatial
frequency in units of *cycles per distance* instead of the much more
common *radians per distance*.) In the image plane at \(z = 0\), there
is no defocus and we get the amplitude point spread function (PSF) as
the Fourier transform of the pupil function \(P \left( k_x, k_y
\right)\) just like we would expect (see Goodman^{2}, Chapter 6,
pp. 129-131).

The problem arises when I try to verify this model when \(z\) is not equal to zero by computing the defocused PSF using the wavefront error for defocus. From the scalar diffraction theory of aberrations, the defocused PSF is the Fourier transform of the pupil function multiplied by a phase factor whose phase angle is proportional to the wavefront error \(W \left( k_x, k_y \right)\)

\begin{equation*} \text{PSF}_{\text{A}} \left( x, y, z \right) = \iint_{pupil} P \left( k_x, k_y \right) e^{i k W \left( k_x, k_y \right)} e^{i \left( k_{x}x + k_{y}y \right)} dk_x dk_y \end{equation*}
Any textbook discussing Seidel aberrations will tell you that the
defocused wavefront error \(W\) is quadratic in the pupil plane
coordinates, i.e. \(W \sim k_x^2 + k_y^2\). Goodman even states this
with little justification later in Chapter 6 on p. 149 when discussing
defocus ^{2}. So how can I reconcile the first equation in which the
defocus goes as the square root of a constant minus the squared pupil
coordinates with the second equation that is quadratic in pupil
coordinates?

The answer, as you might have guessed from the title of this post, is
that *the Seidel polynomial term for defocus is a result of applying
the paraxial approximation when computing the wavefront error*. You
can see this by calculating the phase of a spherical wave in the pupil
plane that is centered on the axis in the image plane; Goodman states
it is quadratic without justification, but this is only true near the
axis. Mahajan offers a geometrical interpretation of \(W\) on p. 148,
where he notes that the path length difference for defocus "is
approximately equal to the difference of sags of the reference sphere
and the wavefront.^{3}" He then goes on to derive the same expression
for \(W\) that Goodman gets; he states the approximation that he uses,
whereas Goodman implicitly assumes a quadratic phase front in the
pupil. Hanser et al., on the other hand, are essentially propagating
the plane wave angular spectrum from the image plane to nearby planes
to model defocus. I think that this should be rigorously correct since
no approximation is applied. For me, it is unfortunate that this was
not made clear in their paper because I spent quite a bit of time
trying to reconcile the two results.

The lesson of this story is to be sure you know about the approximations that go into a "standard" result found in textbooks. I falsely assumed that the Seidel aberrations were exact and that Hanser, et al. were suspect when in fact it was the other way around. Because the paraxial approximation is so widespread in optics theory, it can often lie hidden behind an equation and its effects can easily be taken for granted. This is a problem for the large NA systems used in microscopy where the results of the paraxial approximation are not often valid.

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