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:

\begin{equation*} \text{PSF}_{\text{A}} \left( x, y, z \right) = \iint_{pupil} P \left( k_x, k_y \right) e^{i k_z z} e^{i \left( k_{x}x + k_{y}y \right)} dk_x dk_y \end{equation*}

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 Goodman2, 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.