The lens is one of the most commonly used optical elements. Yet it is sometimes difficult to make accurate effective focal length and pupil position measurements, especially for long focal length lenses. Many measurement methods rely on a mechanical measurement to determine the back focal length, or may require careful operator discrimination in determining the best focus position. Aberrations may confuse an automatic focal length measurement system. However, an accurate determination of the optical properties of a lens is often critical for building an accurate system model. We have developed a method for measurement of the focal length, pupil plane and collimation positions of positive lenses using a Shack-Hartmann wavefront sensor. The SHWFS uses a micro-optic lens array to separate the incoming wavefront into a pattern of focal spots. The position of these focal spots is related to the local wavefront slope. Wavefront reconstruction allows the complete incident wavefront to be retrieved. A Zernike decomposition reconstructor is used to separate the effects of lens focal power from other aberrations. The lens under test is illuminated by a point source on a computer-controlled stage. The transmitted wavefront was recorded by the SHWFS while the source was translated over a few mm range. By analyzing the Zernike coefficient associated with defocus, we were able to extract the focal length, pupil plane and collimation positions using a least squares fitting procedure. This procedure was tested for a variety of lenses of varying focal lengths, from 10 to 1000 mm focal length, and showed excellent repeatability and accuracy. These measurements were compared to knife-edge, manufacturer’s specification, and ray-tracing analysis for verification testing.
The circle polynomials of Zernike are a vital tool in the analysis of optical systems. Decomposition of wavefronts into Zernike polynomials can be insightful. Computation in the Zernike basis, however, is quite cumbersome and inefficient. This paper will address how rational polynomials such as Zernike, Laguerre, Legendre and Chebyshev can be represented as affine combinations of a
Taylor monomial set. This paper also demonstrates the efficiency of common routines using a Taylor basis as well as implementation and optimization issues.
The circle polynomials of Zernike play a prominent role in optical analysis. While decompositions of wavefronts into Zernike polynomial series can yield valuable insight, computing with the polynomials themselves is quite inefficient. Here we outline how rational polynomials like those of Zernike, Legendre, Chebyshev and Laguerre can be handled as affine combinations of a Taylor monomial set. We demonstrate how calculations can be performed much more rapidly in the Taylor basis and how to use integer transformations to recover the exact amplitudes in the desired basis. We also explore C++ optimizations for storing the Zernike amplitudes and transforming between Zernike polynomials and Taylor monomials.