In many applications of freeforms, it is sufficient to choose the domain for orthogonalization to be a circular cylinder that tightly encloses the aperture. In this case, *ρ*_{max} is taken to be the radius of the enclosing cylinder and *N*_{bfs} becomes a function of the polar angle mentioned leading into Eq. (1). It turns out that all the monomial terms of Eq. (2) are accounted for if we now express the normal departure of Eq. (3) in polar coordinates as:

Again, only the lower limits of the indices of summation are shown; much as for the sums in Eq. (2), their upper limits can be chosen in a number of ways. The first line of Eq. (5) contains the *m*=0 terms that are precisely those of Eq. (4).

The equivalence of Eq. (5) and a Cartesian monomial sum like that in Eq. (2) can be appreciated by expanding the right-hand side and then equating real and imaginary parts of the identity:

With this, it can be seen that all terms up to, say, order *T* are included if Eq. (2) is summed over positive indices satisfying *j*+*k*≤*T*, or Eq. (5) is summed over:

Note that the *T*+1 degrees of freedom at order *T* are with *j*=0,1,…*T* and *k*=*T*-*j* for the sum in Eq. (2), whereas, for Eq. (5), they are and where *m* has the same parity (even/odd) as *T* with 0≤*m*≤*T* and

[When counting to identify the *T*+1 terms for any *T*>2, note that does not enter Eq. (5), and this drops one of the terms when *T* is even.] If the rotationally symmetric components of Eq. (5) are supplemented by *u*^{0} and *u*^{2} (i.e., piston and power), all terms up to order *T* in either Eq. (5) or in a Cartesian monomial sum have equal numbers of degrees of freedom and are precisely interchangeable linear combinations of each other.

For any given weighting function, the polynomials in Eq. (5) can be determined uniquely by requiring that the weighted mean square gradient of *N*_{bfs}(*u*,θ) is just the sum of the squares of all the coefficients, see [5]. Although – as shown in the plots in Appendix 1 – the results differ significantly from Zernike polynomials, these basis elements can be labeled in terms that are familiar from the Zernike domain. The four basis members that typically dominate the spectra are plotted in Figure 3are referred to in what follows with familiar labels: astigmatism, coma, trefoil, and spherical aberration. Also, note that, although the tilt terms associated with and may sometimes be significant, they are of minor importance as far as manufacturability is concerned. In fact, they can be taken to be zero without loss of generality provided the part has tip and tilt freedoms when it is configured in the system during optimization.

Figure 3 Plots of the lower-order basis elements for *m*=0, 1, 2, and 3 (spherical, coma, astigmatism, and trefoil, respectively). These are all plotted on the same scale and are of 4th, 3rd, 2nd, and 3rd order with peak values of 0.25, 0.41, 0.71, and 0.54, respectively.

Just as for the rotationally symmetric case, it is instructive to express freeform surfaces from the patent literature in both the representations considered here. One such example is shown in Figure 4 where the listed coefficients from the patent include all terms up to order 10 in Eq. (2) and are in that upper table. Both conic constants vanish for this mirror and the radii in mm are 1/*c*_{x}=-452.62713 and 1/*c*_{y}=-443.43539. This mirror has a plane of symmetry and the coordinates are aligned so that *A*_{jk}=0 whenever *j* is odd. Note that two terms were dropped from the specification, namely (*j,k*)=(0,0) and (0,1), which makes the polynomial and its first derivatives vanish at the origin.

Figure 4 Alternative representations of M6 from ‘projection optics 37’ of US patent no. 2012/0069315 A1 [7]. The Cartesian coefficients from the patent are listed along with the new spectrum of coefficients (to *T*=10) for this shape.

When using Eq. (5), the plane of symmetry means that vanishes for all *m* and *n* when the coordinates are related by (*x*,*y*)=(*u* sin *θ*, *u* cos *θ*). (This choice makes changing the sign on *θ* equivalent to changing the sign on *x* and, just as for the conventional description, all the odd terms then drop out.) With *ρ*_{max}=174.2 and 1/c=-478.12597, the resulting values of are given in the lower table of Figure 4. The illuminated aperture for this part is roughly elliptical with an aspect ratio of 93%, and hence is slightly smaller than the enclosing circle used here. These two characterizations match to better than 1 nm over the enclosing circle, but the second of them requires approximately one-third the number of digits and is plainspoken: this shape is evidently dominated by a couple of hundred microns of each of coma and trefoil . Keep in mind that, as demonstrated in [5], the value of the mean tilt is largely irrelevant to the shape; it relates more to the orientation of the mirror and, much as was done with *A*_{01} in the original specification, it can be dropped from the spectrum provided tip and tilt freedoms are used during optimization. The issue of how easily such an asphere can be produced is discussed briefly in the next subsection.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.