Ordering of terms in different representations of Zernike polynomials

Question

I understand that there are a couple of different representations of Zernike polynomials in general use. What Zernike representation does OSLO use and where?

Synopsis

The ordering of terms in different representations of Zernike polynomials

Solution

Zernike polynomial representation of wavefronts and physical surfaces is widely used in interferometry and optical testing. The Zernike polynomials are one of the infinite numbers of complete sets of polynomials that are orthogonal over the interior of the unit circle. Further discussion of Zernike polynomial analysis can be found in the OSLO 6.1 Optics Reference manual(hardcopy version: pp. 115-119; electronic version pp. 124-128).

There are two popular sets of Zernike polynomials that differ only in the way they are ordered. The Standard set ¹ has an infinite number of terms. The Fringe set ² is a reordered subset of the Standard Zernike terms, with a total of 37 terms. The Fringe set includes higher-order radially symmetric terms while excluding the higher-order azimuthal terms ³.

The Fringe Zernike polynomials are used throughout OSLO for representing wavefront and surface deformations. For a complete listing of the places within OSLO where Fringe Zernike polynomials are used in different OSLO editions, please see the list below. A sample listing of the Fringe Zernike terms can be found in the OSLO 6.1 Optics Reference manual and in the OSLO on-line help:

"Contents>>Lens Data entry>>Special Surface Data>>Diffractive Surfaces>>Zernike Phase"

The Standard Zernike polynomials are only used in OSLO as an option within the interferogram file deformation utility. A sample listing of the Standard Zernike terms can be found in the OSLO on-line help: "Contents>>Lens Data entry>>Special Surface Data>>Interferometric Deformation"

FOOTNOTES:

¹M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1964).

²J.C. Wyant and K. Creath, "Basic Wavefront Aberration Theory for Optical Metrology," pp. 2-53, in Applied Optics and Optical Engineering, Vol. XI, R.R. Shannon and J.C. Wyant, Eds. (Academic Press, Boston, 1992)

³Some of these statements are taken from K. Doyle, V. Genberg, and G. Michels, Integrated Optomechanical Analysis (The Society of Photo-Optical Instrumentation Engineers, SPIE TT58, 2002).

Note:

• (Fringe) Zernike wavefront analysis is available in all editions of OSLO.
• The (Fringe) Symmetric Zernike Sag surface type (an aspheric surface) is available in OSLO Light, Standard and Premium.
• The (Fringe) Asymmetric Zernike Sag surface type (an aspheric surface) is available only in OSLO Premium.
• The (Fringe) Symmetric Zernike Phase surface type (a diffractive surface) is available in OSLO Light, Standard and Premium.
• The (Fringe) Asymmetric Zernike Phase surface type (a diffractive surface) is available only in OSLO Premium.
• An interferogram deformation file (either Fringe or Standard, and either surface or phase deviation) can be applied to surfaces only in OSLO Premium.