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OPTIMIZATION OF ZONAL WAVEFRONT ESTIMATION AND CURVATURE MEASUREMENTS
 Date Issued:
 2007
 Abstract/Description:
 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 gradientbased wavefront sensing, a zonal leastsquares wavefront estimation algorithm for any irregular pupil shape and size is presented, for which the normal matrix equation sets share a predefined matrix. A Gerchberg–Saxton iterative method is employed to reduce the deviation errors in the estimated wavefront caused by the predefined 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 estimationrelated 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 pistonvalue fixed, the oddnumber grid sizes yield lower error propagation than the evennumber grid sizes for all geometries. The Fried geometry either allows subsized wavefront estimations within the testing domain or yields a tworank 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 gradientbased and wavefront differencebased estimations, the Southwell geometry is shown to offer the lowest error propagation with the minimumnorm leastsquares 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 oddnumber grid size. For curvaturebased wavefront sensing, a concept for a differential ShackHartmann (DSH) curvature sensor is proposed. This curvature sensor is derived from the basic ShackHartmann 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 ShackHartmann grid coordinates of the three channels, the differentials of the wavefront slope at each ShackHartmann grid point can be obtained, so the Laplacian curvatures and twist terms will be available. The acquisition of the twist terms using a Hartmannbased 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 leastsquaresbased 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 ShackHartmann slope sensor and Roddier's curvature sensor. It is a twodimensional 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 twodimensional 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 DSHcurvaturesensorbased wavefront estimation does not suffer from the 2ambiguity problem, so potentially both small and large aberrations may be measured.
Title:  OPTIMIZATION OF ZONAL WAVEFRONT ESTIMATION AND CURVATURE MEASUREMENTS. 
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Name(s): 
Zou, Weiyao, Author Rolland, Jannick, Committee Chair University of Central Florida, Degree Grantor 

Type of Resource:  text  
Date Issued:  2007  
Publisher:  University of Central Florida  
Language(s):  English  
Abstract/Description:  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 gradientbased wavefront sensing, a zonal leastsquares wavefront estimation algorithm for any irregular pupil shape and size is presented, for which the normal matrix equation sets share a predefined matrix. A Gerchberg–Saxton iterative method is employed to reduce the deviation errors in the estimated wavefront caused by the predefined 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 estimationrelated 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 pistonvalue fixed, the oddnumber grid sizes yield lower error propagation than the evennumber grid sizes for all geometries. The Fried geometry either allows subsized wavefront estimations within the testing domain or yields a tworank 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 gradientbased and wavefront differencebased estimations, the Southwell geometry is shown to offer the lowest error propagation with the minimumnorm leastsquares 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 oddnumber grid size. For curvaturebased wavefront sensing, a concept for a differential ShackHartmann (DSH) curvature sensor is proposed. This curvature sensor is derived from the basic ShackHartmann 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 ShackHartmann grid coordinates of the three channels, the differentials of the wavefront slope at each ShackHartmann grid point can be obtained, so the Laplacian curvatures and twist terms will be available. The acquisition of the twist terms using a Hartmannbased 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 leastsquaresbased 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 ShackHartmann slope sensor and Roddier's curvature sensor. It is a twodimensional 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 twodimensional 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 DSHcurvaturesensorbased wavefront estimation does not suffer from the 2ambiguity problem, so potentially both small and large aberrations may be measured.  
Identifier:  CFE0001566 (IID), ucf:47145 (fedora)  
Note(s): 
20070501 Ph.D. Optics and Photonics, College of Optics and Photonics Doctorate This record was generated from author submitted information. 

Subject(s): 
wavefront sensing wavefront estimation irregular pupil shape GerchbergSaxton iterations error propagation matrix eigenvalue differential ShackHartmann curvature sensor principal curvatures and directions twist curvature term 

Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0001566  
Restrictions on Access:  private 20070401  
Host Institution:  UCF 