ABSTRACT
In a general manner, without assuming any symmetries, the shape of a wavefront may be represented by the polynomial:
Wðx,yÞ ¼ Xk i¼0
cijx jyij ð7:1Þ
including high-order aberration terms, where k is the degree of this polynomial. In polar coordinates we define
x ¼ S sin y ð7:2Þ
and
y ¼ S cos y ð7:3Þ
where the angle y is measured with respect to the y axis, as shown in Fig. 7.1. Then, the wavefront shape may be written as
WðS,yÞ ¼ Xk n¼0
Snðanl coslyþ bnl sinlyÞ ð7:4Þ
where the cos y and sin y terms describe the symmetrical and antisymmetrical components of the wavefront, respectively. However, not all possible values of n and l are permitted. To have a single valued function we must satisfy the condition:
WðS,yÞ ¼WðS,yþ pÞ ð7:5Þ
Then, it is easy to see that n and l must both be odd or both even. If this expression for the wavefront is converted into cartesian coordinatesW(x, y),
it becomes an infinite series, unless l n. Thus, if we want Eq. (7.4) to be equivalent to the finite series in Eq. (7.1), we impose this condition, that is almost always satisfied, except in some very rare cases related to rotational shearing interferograms, as pointed out by Malacara and DeVore (1992).
