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.
A problem with wavefront reconstruction is that smaller details are often smeared out in the background of larger noise. A least squares fit tries to distribute measurement errors over the entire region. Features of the size of the RMS error tend to get obscured. We present a reconstruction algorithm and framework that redresses some of these problems. Small patches are reconstructed and then joined together. We discuss how this is implemented.
We present a high-speed silicon wafer metrology tool capable of resolving surface features in the nanometer height range. This tool uses a high performance Shack-Hartman sensor to analyze the wavefront of a beam of light reflected from a silicon wafer surface. By translating the wafer to analyze small portions of the wafer in each camera frame and then continuously piecing the frames together, we can retain sub-millimeter spatial resolution while rapidly analyzing large apertures. This tool is particularly effective for resolving features near the wafer edge. We will describe the measures required to obtain this level of resolution. We also compare data taken with this device to that taken with the National Institute of Standards and Technology X-ray optics Calibration Interferometer (XCALIBIR). Finally, we show measurements of a variety of typical and atypical 200 mm diameter wafer samples.
In general the center of mass technique is a fast and robust way to approximate the location of a focal spot. This paper discusses in detail the relationship between the focal spot location and the center of mass. We start with a mathematical analysis and conclude with a few practical ideas to improve the accuracy of the center of mass technique.
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.
The Shack-Hartmann wavefront sensor is a powerful tool for optical analysis. A vital step in the data analysis chain involves reconstructing the wavefront incident upon the lenslet array from hundreds or thousands of slope measurements. We discuss here the basics of reconstruction and the differences in reconstruction for the case of Hartmann and Shack-Hartmann sensors. Hartmann sensors take point samples of the wavefront but Shack-Hartmann sensors measure the average slope over small regions. This results in subtle differences in the reconstruction.
A problem with wavefront reconstruction is that smaller details are often smeared out in the background of larger noise. A least squares fit tries to distribute measurement errors over the entire region. Features of the size of the RMS error tend to get obscured. We present a reconstruction algorithm and framework that redresses some of these problems. Small patches are reconstructed and then joined together. We discuss how this is implemented.
The most critical element in a ocular Shack-Hartmann wavefront sensor is the micro-optic lenslet array. This array largely determines the accuracy of the wavefront measurement and the dynamic range of the measurements. This paper discusses the details of how the density of the lenslet array affects the accuracy of the wavefront measurement. We briefly discuss wavefront reconstruction, which is the mathematical process that takes the output from the lenslet array and reconstructs the input wavefront. We compare the two primary methods of reconstruction, the zonal fit and the modal fit. We also show how a denser array can be designed to have a better dynamic range.
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