A Shack-Hartmann wavefront sensor (SHWS) consists of a microlens array mounted in front of a CCD array. Such a device can measure the wavefront characteristics of a light beam, its transmission through an optical element, or its reflection off an optical surface such as a lens or mirror. SHWSs are deceptively simple; in reality, successful design and implementation require a number of engineering tradeoffs.
The individual microlenses sample the beam under test to create a series of focal spots on a finely spaced CCD array. The centroids of the spots are calculated by determining their center-of-mass coordinates. The test process usually involves comparing the centroid positions for the focal spot of the beam under test with a file of the centroid positions for a flat reference wavefront. This reference file is used to minimize the effects from imperfections in the CCD array, microlens array, and imaging system. We can calculate the average gradient of the wavefront over each microlens from the difference between the centroid positions of the test beam and those of the reference wavefront, which can be integrated to reconstruct the incident wavefront.
Figure 1. The results of SHWS with different microlens array focal lengths demonstrate the tradeoff between RMS accuracy and dynamic range.
When designing a wavefront sensor, it is important to understand how the microlens array affects the performance of the SHWS. The primary specifications for an SHWS include the spatial resolution (number of microlenses across the CCD area), the wavefront measure- ment accuracy, and the dynamic range, which is related to the maximum wavefront tilt that can be measured.1 The tradeoff between these specifications in an SHWS is mainly determined by the focal length and width of the microlenses in the array,2 in addition to the size of the individual pixels of the CCD.
The microlens array of an SHWS is normally placed a focal distance away from the CCD array. Nominally, a shorter focal length microlens will yield a larger dynamic range but at the cost of reduced wavefront measurement accuracy. Essentially this accuracy is governed by the size of the focal spot, since a sufficient number of pixels must be illuminated to ensure accurate calculation of the centroid position. For a square microlens with a 100% fill factor, the FWHM spot size S is:
S = 2fλ/d
where λ is the wavelength of the incident light, f is the focal length, and d is the width of the microlens. For optimal performance, each focal spot should illuminate a patch of at least 3 x 3 pixels; in general, smaller pixels yield better accuracy.
In the small angle approximation, the dynamic range D per microlens is given by
D = (d/2 - S/2)2/f
The total dynamic range of the sensor is ND, where N is the number of lenslets across the array.
The tradeoffs between spatial resolution, dynamic range, and wavefront measurement accuracy impose certain limits on the design of the microlens array. For example, a low f/d ratio creates too small of a focal spot, compromising accuracy. Increasing width enhances dynamic range but at the cost of decreased spatial resolution; moreover, if spots are too large (i.e., longer focal lengths or smaller widths), focal spots from adjacent microlenses will overlap, reducing the accuracy of the SHWS.
Wavefront accuracy measurements tested with an SHWS in conjunction with their respective dynamic ranges are shown in the figure for a series of microlens arrays with different focal lengths and widths. One can clearly see the tradeoffs for the different microlens arrays. As the focal length increases, the dynamic range decreases, and the accuracy increases at the expense of reduced spatial resolution. For given accuracy and dynamic range requirements, this comparison of SHWS performance with different types of microlenses reveals the tradeoffs that should be taken into consideration when designing SHWS microlens arrays. oe
1. D. Neal, et al., Proc. SPIE 4779, pp. 148-160 (2002).
2. P. Pulaski, et al., Proc. SPIE 4767, pp.44-52 (2002).
Paul Pulaski is optical scientist and product manager at WaveFront Sciences Inc., Albuquerque, NM.