On a propagation-invariant, orthogonal modal expansion on the unit disk: going beyond Nijboer-Zernike theory of...

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.