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Nijboer–Zernike’s circle polynomials are broadly used for the evaluation of aberrations of optical systems or, more generally, wavefront analysis. This is because they are orthogonal over a unit circle and are directly related to the balanced classical aberrations for imaging systems with circular pupils. However, such expansion and successive extensions of the original theory suffer from a key limitation: it does not preserve its form under propagation. This means that even if a Nijboer–Zernike expansion for a field is known on a given reference plane, as soon as another plane is considered, a new, different set of polynomials for the same field appears. The origin of this problem is to be ascribed to the fact that Nijboer–Zernike polynomials are a useful mathematical tool which, however, are not bound to the physics of the electromagnetic problem they are employed in. In this work, we show that a more appropriate modal expansion does exist that is not only orthogonal on the unit disk but is also invariant on propagation. Besides impacting the field of aberrations analysis and retrieval, the modal expansion introduced here holds an enormous potential for digital classical and quantum optical communications, optical metrology, and adaptive optics too. The practical implementation, physical interpretation, and visualization of this new modal expansion are all very straightforward.

We introduce the degree of paraxiality as a paraxiality measure of a monochromatic light beam. Computation of this parameter is possible once the plane-wave spectrum of the field being analyzed is given for a starting source plane. On the basis of this definition, quantitative comparisons of the paraxiality of different types of fields are possible. In addition, the recently introduced paraxial estimator [Opt. Lett.32, 927 (2007)] is identified as the limiting case of the degree of paraxiality.

In our recently published Reply [Opt. Lett. 33, 3006 (2008)], we mistakenly wrote the variable βλ instead of the correct βw0.

The theory of the degree of paraxiality is correct. Very likely, the Comment [Opt. Lett.33, 3004 (2008)] on our original Letter [Opt. Lett.33, 1360 (2008)] originated from an imprecision we made in the plot included in the paper. Such imprecision induced the Comment's author to raise further objections that, as proved in the present Reply, are totally incorrect.

A linearly polarized optical field can be obtained by filtering a stochastic field through an ideal linear polarizer. The produced field possesses a given degree of paraxiality that, as proved in the present Letter, can be affected by the correlations of the original stochastic field. An example with Gaussian beams is discussed in detail.