wavefront aberrations clearly map onto specific Zernike polynomials. For example, spherical aberration has no polar angle dependence, but does have a fourth order dependence upon pupil function. This suggests that this aberration has a radial order, n, of 4 and a polar dependence, m, of zero. Similarly, coma has a radial order of 3 and a polar dependence of one. Table 5.2 provides a list of the first 28 Zernike polynomials.
In Table 5.2, each Zernike polynomial has been assigned a unique number. This is the ‘Standard’ numbering convention adopted by the American National Standards Institute, (ANSI). It has the benefit of following the Born and Wolf notation logically, starting from the piston term which is denominated the zeroth term. If the ANSI number is represented as j, and the Born and Wolf indices as n, m, then the ANSI number may be derived as follows:
(5.22)
Unfortunately, a variety of different numbering conventions prevail, leading to significant confusion. This will be explored a little later in this chapter. As a consequence of this, the reader is advised to be cautious in applying any single digit numbering convention to Zernike polynomials. By contrast, the n, m numbering convention used by Born and Wolf is unambiguous and should be used where there is any possibility of confusion.
5.3.3 Zernike Polynomials and Aberration
As outlined previously, there is a strong connection between Zernike polynomials and primary aberrations when expressed in terms of wavefront error. Table 5.2 clearly shows the correspondence between the polynomials and the Gauss Seidel aberrations, with the 3rd order Gauss-Seidel aberrations, such as spherical aberration and coma clearly visible.
The power of the Zernike polynomials, as an orthonormal set, lies in their ability to represent any arbitrary wavefront aberration. Using the approach set out in Eq. (5.13), it is possible to compute the magnitude of any Zernike term by the cross integral of the relevant polynomial and the wavefront disturbance. Furthermore, the total root mean square (rms) wavefront error, as per Eq. (5.14), may be calculated from the RSS (root sum square) of the individual Zernike magnitudes. That is to say, the Zernike magnitude of each term represents its contribution to the rms wavefront error, as averaged over the whole pupil.
The use of defocus to compensate spherical aberration was explored in Chapters 3 and 4. In this instance, for a given amount of fourth order wavefront error, we sought to minimise the rms wavefront error by applying a small amount of defocus.
Hence, without defocus, adjustment, the raw spherical aberration produced in a system may be expressed as the sum of three Zernike terms, one spherical aberration, one defocus and one piston term. The total aberration for an uncompensated system is simply given by the RSS of the individual terms. However, for a compensated system only the Zernike n = 4, m = 0 term needs be considered. This then gives the following fundamental relationship:
Table 5.2 First 28 Zernike polynomials.
| ANSI# | N | m | Nn,m | R(ρ) | G(ϕ) | Name |
| 0 | 0 | 0 | 1 | 1 | 1 | Piston |
| 1 | 1 | −1 |
|
ρ | sin φ | Tilt X |
| 2 | 1 | 1 |
|
ρ | cos φ | Tilt Y |
| 3 | 2 | −2 |
|
ρ 2 | sin 2φ | 45° Astigmatism |
| 4 | 2 | 0 |
|
2ρ2 − 1 | 1 | Defocus |
| 5 | 2 | 2 |
|
ρ 2 | cos 2φ | 90° Astigmatism |
| 6 | 3 | −3 |
|
ρ 3 | sin 3φ | Trefoil |
| 7 | 3 | −1 |
|
3ρ3 − 2ρ | sin φ | Coma Y |
| 8 | 3 | 1 |
|
3ρ3 − 2ρ | cos φ | Coma X |
| 9 |
3
|
