Sag- and slope-orthogonal bases to characterise nominally rectangular freeform surfaces

Sag- or slope-orthogonal bases have proven effective for characterising aspheric and freeform surfaces on circular apertures. Things are not so straightforward, however, for rectangular apertures when slope orthogonality is desired. While such a basis can be constructed via Gram-Schmidt, we show that the simplifying step offered by separation of variables fails, the basis depends on the domain's aspect ratio, and no recurrence relations exist for efficient computation. In contrast, a characterisation in terms of an RDF basis [Opt. Express27, 32263 (2019)10.1364/OE.27.032263] avoids all three issues. Further, the result is both sag- and slope-orthogonal making it ideal for on-the-fly design constraints.