wavefront sensing, wavefront estimation, irregular pupil shape, Gerchberg-Saxton iterations, error propagation, matrix eigenvalue, differential Shack-Hartmann curvature sensor, principal curvatures and directions, twist curvature term


Optical testing in adverse environments, ophthalmology and applications where characterization by curvature is leveraged all have a common goal: accurately estimate wavefront shape. This dissertation investigates wavefront sensing techniques as applied to optical testing based on gradient and curvature measurements. Wavefront sensing involves the ability to accurately estimate shape over any aperture geometry, which requires establishing a sampling grid and estimation scheme, quantifying estimation errors caused by measurement noise propagation, and designing an instrument with sufficient accuracy and sensitivity for the application. Starting with gradient-based wavefront sensing, a zonal least-squares wavefront estimation algorithm for any irregular pupil shape and size is presented, for which the normal matrix equation sets share a pre-defined matrix. A Gerchberg–Saxton iterative method is employed to reduce the deviation errors in the estimated wavefront caused by the pre-defined matrix across discontinuous boundary. The results show that the RMS deviation error of the estimated wavefront from the original wavefront can be less than λ/130~ λ/150 (for λ equals 632.8nm) after about twelve iterations and less than λ/100 after as few as four iterations. The presented approach to handling irregular pupil shapes applies equally well to wavefront estimation from curvature data. A defining characteristic for a wavefront estimation algorithm is its error propagation behavior. The error propagation coefficient can be formulated as a function of the eigenvalues of the wavefront estimation-related matrices, and such functions are established for each of the basic estimation geometries (i.e. Fried, Hudgin and Southwell) with a serial numbering scheme, where a square sampling grid array is sequentially indexed row by row. The results show that with the wavefront piston-value fixed, the odd-number grid sizes yield lower error propagation than the even-number grid sizes for all geometries. The Fried geometry either allows sub-sized wavefront estimations within the testing domain or yields a two-rank deficient estimation matrix over the full aperture; but the latter usually suffers from high error propagation and the waffle mode problem. Hudgin geometry offers an error propagator between those of the Southwell and the Fried geometries. For both wavefront gradient-based and wavefront difference-based estimations, the Southwell geometry is shown to offer the lowest error propagation with the minimum-norm least-squares solution. Noll’s theoretical result, which was extensively used as a reference in the previous literature for error propagation estimate, corresponds to the Southwell geometry with an odd-number grid size. For curvature-based wavefront sensing, a concept for a differential Shack-Hartmann (DSH) curvature sensor is proposed. This curvature sensor is derived from the basic Shack-Hartmann sensor with the collimated beam split into three output channels, along each of which a lenslet array is located. Three Hartmann grid arrays are generated by three lenslet arrays. Two of the lenslets shear in two perpendicular directions relative to the third one. By quantitatively comparing the Shack-Hartmann grid coordinates of the three channels, the differentials of the wavefront slope at each Shack-Hartmann grid point can be obtained, so the Laplacian curvatures and twist terms will be available. The acquisition of the twist terms using a Hartmann-based sensor allows us to uniquely determine the principal curvatures and directions more accurately than prior methods. Measurement of local curvatures as opposed to slopes is unique because curvature is intrinsic to the wavefront under test, and it is an absolute as opposed to a relative measurement. A zonal least-squares-based wavefront estimation algorithm was developed to estimate the wavefront shape from the Laplacian curvature data, and validated. An implementation of the DSH curvature sensor is proposed and an experimental system for this implementation was initiated. The DSH curvature sensor shares the important features of both the Shack-Hartmann slope sensor and Roddier’s curvature sensor. It is a two-dimensional parallel curvature sensor. Because it is a curvature sensor, it provides absolute measurements which are thus insensitive to vibrations, tip/tilts, and whole body movements. Because it is a two-dimensional sensor, it does not suffer from other sources of errors, such as scanning noise. Combined with sufficient sampling and a zonal wavefront estimation algorithm, both low and mid frequencies of the wavefront may be recovered. Notice that the DSH curvature sensor operates at the pupil of the system under test, therefore the difficulty associated with operation close to the caustic zone is avoided. Finally, the DSH-curvature-sensor-based wavefront estimation does not suffer from the 2π-ambiguity problem, so potentially both small and large aberrations may be measured.


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Graduation Date





Rolland, Jannick


Doctor of Philosophy (Ph.D.)


College of Optics and Photonics


Optics and Photonics

Degree Program









Release Date

May 2007

Length of Campus-only Access


Access Status

Doctoral Dissertation (Open Access)