While a wavefront represents a surface of constant phase and an aberration function
represents its deviation from a spherical surface, called the reference sphere, we will use
the two terms, wavefront and aberration function, synonymously. In the previous
chapters, we have emphasized that for wavefront analysis, i.e., to determine the content
of an aberration function, a set of polynomials that are orthonormal over the pupil of an
imaging system and represent balanced classical aberrations for the system must be used.
The utility of using the orthonormal polynomials, as opposed to the orthogonal, is that
each expansion coefficient is not only independent of the number of polynomials used in
the expansion, but also represents the standard deviation or the sigma value of the
corresponding polynomial aberration term (with the exception of piston). The variance of
the aberration function is then simply equal to the sum of the squares of the aberration
In this chapter, we consider how best to determine the orthonormal expansion or the
aberration coefficients from the wavefront data measured at an array of points, as, for
example, in a phase-measuring interferometer. The problem of determining the
expansion coefficients when the measured data are the wavefront slopes, as, for example,
in a Shack-Hartmann sensor is also discussed. Although we have considered optical
imaging systems with several different pupil shapes, our focus in this chapter is on a
system with a circular pupil. The analysis given here for such a pupil can be extended to
systems with other pupil shapes.
In practice, what is needed in both optical design and fabrication is the wavefront.
The wavefront aberrations determine the image quality in optical design. In fabrication
and testing of an optical surface, the wavefront errors determine surface errors, and thus
the polishing requirements to obtain the desired surface. Similarly, in adaptive optics, the
signal for the actuators of a deformable mirror to negate the aberrations, such as those
introduced by atmospheric turbulence, comes from the wavefront data. Hence, there is a
need to determine the Zernike coefficients from the wavefront data measured by a
wavefront sensor, or from the slope data provided by a slope sensor. In this chapter, we
present the two main mathematical approaches to determine the expansion coefficients:
an integration method for orthogonal solutions and the classic least squares approach.
We also illustrate the methods with some numerical examples for determining the
Zernike coefficients from the wavefront data or the wavefront slope data. The key points
considered are: how the number of data points affects the accuracy of the coefficients, how the noise in the data affects this accuracy, and how many Zernike polynomials are
needed for adequate representation of the data.