Wavefront expansion basis functions and their relationships

Comparison of Low Degree/High Degree and Zernike Expansions for Evaluating Simulation Outcomes After Customized Aspheric Laser Corrections

Purpose

The purpose of this study was to compare the low degree/high degree (LD/HD) and Zernike Expansion simulation outcomes evaluating the corneal wavefront changes after theoretical conventional and customized aspheric photorefractive ablations.

Methods

Initial anterior corneal surface profiles were modeled as conic sections with pre-operative apical curvature, R0, and asphericity, Q0. Postoperative apical curvature, R1, was computed from intended defocus correction, D, diameter zone, S, and target postoperative asphericity, Q1. Coefficients of both Zernike and LD/HD polynomial expansions of the rotationally symmetrical corneal profile were computed using scalar products. We modeled different values of D, R0, Q0, S, and ΔQ = Q1 to Q0. The corresponding postoperative changes in defocus (Δz20 vs. Δg20), fourth order (Δz40 vs. Δg40) and sixth order (Δz60 vs. Δg60) Zernike and LD/HD spherical aberrations (SAs) were compared. In addition, retrospective clinical data and wavefront measurements were obtained from two examples of two patient eyes before and after corneal laser photoablation.

Results

The z20, varied with both R0 and Q0, whereas the LD/HD defocus coefficient, g20, was relatively robust to changes in asphericity. Variations of apical curvature better correlated with defocus and ΔQ with SA coefficients in the LD/HD classification. The impact of ΔQ was null on g20 but induced significant linear variations in z20 and fourth order SA coefficients. LD/HD coefficients provided a good correlation with the visual performances of the operated eyes.

Conclusions

Simulated variations in postoperative corneal profile and wavefront expansion using the LD/HD approach showed good correlations between defocus and asphericity variations with variations in corneal curvature and SA coefficients, respectively.

Translational Relevance

The relevance of this study was to provide a clinically relevant alternative to Zernike polynomials for the interpretation of wavefront changes after customized aspheric corrections.

Interaction of axial and oblique astigmatism in theoretical and physical eye models

The interaction between oblique and axial astigmatism was investigated analytically (generalized Coddington's equations) and numerically (ray tracing) for a theoretical eye model with a single refracting surface. A linear vector-summation rule for power vector descriptions of axial and oblique astigmatism was found to account for their interaction over the central 90° diameter of the visual field. This linear summation rule was further validated experimentally using a physical eye model measured with a laboratory scanning aberrometer. We then used the linear summation rule to evaluate the relative contributions of axial and oblique astigmatism to the total astigmatism measured across the central visual field. In the central visual field, axial astigmatism dominates because the oblique astigmatism is negligible near the optical axis. At intermediate eccentricities, axial and oblique astigmatism may have equal magnitude but orthogonal axes, which nullifies total astigmatism at two locations in the visual field. At more peripheral locations, oblique astigmatism dominates axial astigmatism, and the axes of total astigmatism become radially oriented, which is a trait of oblique astigmatism. When eccentricity is specified relative to a foveal line-of-sight that is displaced from the eye's optical axis, asymmetries in the visual field map of total astigmatism can be used to locate the optical axis empirically and to estimate the relative contributions of axial and oblique astigmatism at any retinal location, including the fovea. We anticipate the linear summation rule will benefit many topics in vision science (e.g., peripheral correction, emmetropization, meridional amblyopia) by providing improved understanding of how axial and oblique astigmatism interact to produce net astigmatism.

Step-along power vector method for astigmatic wavefront propagation

PURPOSE

To propose both a new algebraic solution and a graphical monitoring method for astigmatic wavefront propagation in the framework provided by power vectors.

METHODS

The generalised propagation equation describing the propagation of astigmatic wavefronts from one plane to another is adapted to the power vectors formalism using a novel algorithm based on a step-along method. The step-along procedure is directly applied to the tuple of power vectors [M, J0 , J45 ] representing an arbitrary astigmatic wavefront and it permits the calculation of the tuple of power vectors [M', J'0 , J'45 ] after a given propagation distance. This is achieved mathematically first by temporarily rotating the astigmatic wavefront so that one of the principal meridians is horizontal, then propagating the wavefront, and finally rotating the propagated wavefront back to its original orientation.

RESULTS

A transfer rule for power vectors representing astigmatic wavefronts is analytically obtained. The new algorithm provides an algebraic solution for the propagation of astigmatic wavefronts using power vectors in a homogeneous medium. In addition, the new step-along procedure allows 2D as well as 3D graphical monitoring of the astigmatic wavefront being referred to the X-Y conventional reference framework by virtue of the trajectories of the power vector coordinates. The proposed solution has been validated through several numerical examples.

CONCLUSIONS

A new step-along method for astigmatic wavefront propagation using power vectors has been presented and validated for classical as well as new numerical examples. The method provides algebraic calculation as well as graphical monitoring of the wavefront vergence during propagation.

Derivation of the propagation equations for higher order aberrations of local wavefronts

From the literature the analytical calculation of local power and astigmatism of a wavefront after refraction and propagation is well known; it is, e.g., performed by the Coddington equation for refraction and the classical vertex correction formula for propagation. Recently the authors succeeded in extending the Coddington equation to higher order aberrations (HOA). However, equivalent analytical propagation equations for HOA do not exist. Since HOA play an increasingly important role in many fields of optics, e.g., ophthalmic optics, it is the purpose of this study to extend the propagation equations of power and astigmatism to the case of HOA (e.g., coma and spherical aberration). This is achieved by local power series expansions. In summary, with the results presented here, it is now possible to calculate analytically the aberrations of a propagated wavefront directly from the aberrations of the original wavefront containing both low-order and high-order aberrations.

Aberrations and spherocylindrical powers within subapertures of freeform surfaces

A method is described for the derivation of refractive properties and aberration structure of subapertures of freeform surfaces. Surface shapes are described in terms of Zernike polynomials. The method utilizes matrices to transform between Zernike and Taylor coefficients. Expression as a Taylor series facilitates the translation and size rescaling of subapertures of the surface. An example operation using a progressive addition lens surface illustrates the method.

Modal estimation of wavefront phase from slopes over elliptical pupils

PURPOSE

Measuring the off-axis optical quality of the eye with a Shack-Hartmann wavefront sensor requires methods for reconstructing wavefront from the gradient data defined within an elliptical pupil. Such methods for modal estimation of wavefront aberrations are sensitive to pupil shape.

METHODS

We develop a conceptual framework that reconciles two published, but apparently dissimilar, methods for reconstruction over an elliptical pupil based on Zernike analysis. Our unified treatment shows that the two methods have different interpretations but the vectors of Zernike coefficients they produce are related linearly. Two novel methods based on Fourier series are also introduced for a model of gradient sensors based on Southwell geometry.

RESULTS

All four methods were evaluated numerically with three test-cases: a defocus wavefront (1), a spherocylindrical wavefront (2), and a random-generated wavefront (3). Under noise-free conditions, all four methods reconstructed the tested wavefronts accurately. The reconstruction error is negligible at the level of numerical computation. Furthermore, the Monte-Carlo simulation with test case 2 revealed small differences in sensitivity to noise between the two Zernike methods but no difference between the two Fourier methods. Because of the smoothing effects, the two Zernike-based methods are more robust to noise than are the two Fourier methods. However, Fourier methods are computationally faster.

CONCLUSIONS

All four modal methods are validated methods to reconstruct wavefronts from the gradients over the elliptical pupil. The choice of these methods is application dependent.

Pupil tracking with a Hartmann-Shack wavefront sensor

We present the theoretical background and experimental validation of a pupil tracking method based on measurement of the irradiance centroid of Hartmann-Shack aberrometric images. The experimental setup consists of a Hartmann-Shack (HS) sensor forming over the same camera the images of the eye's pupil and the aberrometric image. The calibration is made by comparing the controlled displacements induced to an artificial eye with the displacements estimated from the centroid of the pupil and of the HS focal plane. The pupil image is also used for validation of the method when operating with human eyes. The experimental results after calibration show a root mean square error of 10.45 mum for the artificial eye and 27, 10, and 6 mum rms for human eyes tested using Hartmann-Shack images, with signal-to-noise ratios of 6, 8, and 11, respectively. The performance of the method is similar to conventional commercial eye trackers. It avoids the need for using separate tracking devices and their associated synchronization problems. This technique can also be used to reprocess present and stored sets of Hartmann-Shack aberrometric images to estimate the ocular movements that occurred during the measurement runs, and, if convenient, to correct the measured aberrations from their influence.

Bilinear wavefront transformation

Truncated expansions such as Zernike polynomials provide a powerful approach for describing wavefront data. However, many simple calculations with data in this form can require significant computational effort. Important examples include recentering, renormalizing, and translating the wavefront data. This paper describes a technique whereby these operations and many others can be performed with a simple matrix approach using monomials. The technique may be applied to other expansions by reordering the data and applying transformations. The key is the use of the vectorization operator to convert data between vector and matrix descriptions. With this conversion, one-dimensional polynomial techniques may be employed to perform separable operations. Examples are also given for differentiation and integration of wavefronts.

Wavefront propagation from one plane to another with the use of Zernike polynomials and Taylor monomials

In wavefront-driven vision correction, ocular aberrations are often measured on the pupil plane and the correction is applied on a different plane. The problem with this practice is that any changes undergone by the wavefront as it propagates between planes are not currently included in devising customized vision correction. With some valid approximations, we have developed an analytical foundation based on geometric optics in which Zernike polynomials are used to characterize the propagation of the wavefront from one plane to another. Both the boundary and the magnitude of the wavefront change after the propagation. Taylor monomials were used to realize the propagation because of their simple form for this purpose. The method we developed to identify changes in low-order aberrations was verified with the classical vertex correction formula. The method we developed to identify changes in high-order aberrations was verified with ZEMAX ray-tracing software. Although the method may not be valid for highly irregular wavefronts and it was only proven for wavefronts with low-order or high-order aberrations, our analysis showed that changes in the propagating wavefront are significant and should, therefore, be included in calculating vision correction. This new approach could be of major significance in calculating wavefront-driven vision correction whether by refractive surgery, contact lenses, intraocular lenses, or spectacles.

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.

Comparison of Wavefront Reconstructions With Zernike Polynomials and Fourier Transforms

PURPOSE

To make a direct comparison between Fourier and Zernike reconstructions of ocular wavefronts using a newly available analytical theory by which Fourier coefficients can be converted to Zernike coefficients and vice versa.

METHODS

Noise-free random wavefronts were simulated with up to the 15th order of Zernike polynomials. For each case, 100 random wavefronts were simulated separately. These wavefronts were smoothed with a low-pass Gaussian filter to remove edge effects. Wavefront slopes were calculated, and normally distributed random noise was added within the circular area to simulate realistic Shack-Hartmann spot patterns. Three wavefront reconstruction methods were performed. The wavefront surface error was calculated as the percentage of the input wavefront root mean square.

RESULTS

Fourier full reconstruction was more accurate than Zernike reconstruction from the 6th to the 10th orders for low-to-moderate noise levels. Fourier reconstruction was found to be approximately 100 times faster than Zernike reconstruction. Fourier reconstruction always makes optimal use of information. For Zernike reconstruction, however, the optimal number of orders must be chosen manually. The optimal Zernike order for Zernike reconstruction is lower for smaller pupils than larger pupils.

CONCLUSIONS

Fourier full reconstruction is faster and more accurate than Zernike reconstruction, makes optimal use of slope information, and better represents ocular aberrations of highly aberrated eyes.