The statistics of refractive error maps : managing wavefront aberration analysis without Zernike polynomials

The refractive error of a human eye varies across the pupil and therefore may be treated as a random variable. The probability distribution of this random variable provides a means for assessing the main refractive properties of the eye without the necessity of traditional functional representation of wavefront aberrations. To demonstrate this approach, the statistical properties of refractive error maps are investigated. Closed-form expressions are derived for the probability density function (PDF) and its statistical moments for the general case of rotationally-symmetric aberrations. A closed-form expression for a PDF for a general non-rotationally symmetric wavefront aberration is difficult to derive. However, for specific cases, such as astigmatism, a closed-form expression of the PDF can be obtained. Further, interpretation of the distribution of the refractive error map as well as its moments is provided for a range of wavefront aberrations measured in real eyes. These are evaluated using a kernel density and sample moments estimators. It is concluded that the refractive error domain allows non-functional analysis of wavefront aberrations based on simple statistics in the form of its sample moments. Clinicians may find this approach to wavefront analysis easier to interpret due to the clinical familiarity and intuitive appeal of refractive error maps.

Depth of focus and visual acuity with primary and secondary spherical aberration

It is known that the depth of focus (DOF) of the human eye can be affected by the higher order aberrations. We estimated the optimal combinations of primary and secondary Zernike spherical aberration to expand the DOF and evaluated their efficiency in real eyes using an adaptive optics system. The ratio between increased DOF and loss of visual acuity was used as the performance indicator. The results indicate that primary or secondary spherical aberration alone shows similar effectiveness in extending the DOF. However, combinations of primary and secondary spherical aberration with different signs provide better efficiency for expanding the DOF. This finding suggests that the optimal combinations of primary and secondary spherical aberration may be useful in the design of optical presbyopic corrections. © 2011 Elsevier Ltd. All rights reserved.

Multiple imaging with an aberration optimized hololens array

The imaging performance of hololenses formed with four different geometries were studied through an analysis of their third-order aberration coefficients. It is found that the geometry proposed by Brandt (1969) gives the least residual aberration with minimum variation of this aberration with the reconstruction angle. When the ideal position of one of the construction beams is changed in order to generate a hololens array, the residual aberration is found to increase sharply, which in turn affects the image resolution among the multiplied images in the output. A hololens array was generated using Brandt's geometry with the help of a one-dimensional sinusoidal grating. The results of multiple imaging with the hololens array are presented. The image resolution is reasonably high and can be further improved by reducing the f-number of the hololenses.

Spherical aberration and its correction using Lie algebraic methods

We discuss three methods to correct spherical aberration for a point to point imaging system. First, results obtained using Fermat's principle and the ray tracing method are described briefly. Next, we obtain solutions using Lie algebraic techniques. Even though one cannot always obtain analytical results using this method, it is often more powerful than the first method. The result obtained with this approach is compared and found to agree with the exact result of the first method.

Investigation of the influence of the aberration induced by a plane interface on STED microscopy

The structure of the inhibition patterns is important to the stimulated emission depletion (STED) microscopy. Usually, Laguerre-Gaussian (LG) beam and the central zero-intensity patterns created by inserting phase masks in Gaussian beams are used as the erase beam in STED microscopy. Aberration is generated when focusing beams through an interface between the media of the mismatched refractive indices. By use of the vectorial integral, the effects of such aberration on the shape of depletion patterns and the size of fluorescence emission spot in the STED microscopy are studied. Results are presented as a comparison between the aberration-free case and the aberrated cases. (C) 2009 Optical Society of America

Aberration measurement of projection optics in lithographic tools based on two-beam interference theory

The degradation of image quality caused by aberrations of projection optics in lithographic tools is a serious problem in optical lithography. We propose what we believe to be a novel technique for measuring aberrations of projection optics based on two-beam interference theory. By utilizing the partial coherent imaging theory, a novel model that accurately characterizes the relative image displacement of a fine grating pattern to a large pattern induced by aberrations is derived. Both even and odd aberrations are extracted independently from the relative image displacements of the printed patterns by two-beam interference imaging of the zeroth and positive first orders. The simulation results show that by using this technique we can measure the aberrations present in the lithographic tool with higher accuracy. (c) 2006 Optical Society of America.

Aberration measurement of projection optics in lithographic tools by use of an alternating phase-shifting mask

As a critical dimension shrinks, the degradation in image quality caused by wavefront aberrations of projection optics in lithographic tools becomes a serious problem. It is necessary to establish a technique for a fast and accurate in situ aberration measurement. We introduce what we believe to be a novel technique for characterizing the aberrations of projection optics by using an alternating phase-shifting mask. The even aberrations, such as spherical aberration and astigmatism, and the odd aberrations, such as coma, are extracted from focus shifts and image displacements of the phase-shifted pattern, respectively. The focus shifts and the image displacements are measured by a transmission image sensor. The simulation results show that, compared with the accuracy of the previous straightforward measurement technique, the accuracy of the coma measurement increases by more than 30% and the accuracy of the spherical-aberration measurement increases by approximately 20%. (c) 2006 Optical Society of America.

A novel method to measure coma aberration of projection system

A novel method to measure coma aberration by pattern displacements at different defocus positions is proposed in this paper. The effect of defocus on coma-induced pattern displacement is analyzed. The measuring principle of the method is described in detail. Using the simulation program PROLITH, the proportionality factors between pattern displacement and coma aberration at different defocus positions are calculated. It is proved that the method is simple to perform and the measurement accuracy of coma can increase approximately by 25% by this novel method. (c) 2006 Elsevier GmbH. All rights reserved.

Even aberration measurement of lithographic projection system based on optimized phase-shifting marks

In the present paper, we propose a novel method for measuring the even aberrations of lithographic projection optics by use of optimized phase-shifting marks on the test mask. The line/space ratio of the phase-shifting marks is optimized to obtain the maximum sensitivities of Zernike coefficients corresponding to even aberrations. Spherical aberration and astigmatism can be calculated from the focus shifts of phase-shifting gratings oriented at 0 degrees, 45 degrees, 90 degrees and 135 degrees at multiple illumination settings. The PROLITH simulation results show that, the measurement accuracy of spherical aberration and astigmatism obviously increase, after the optimization of the measurement mark. (C) 2008 Elsevier B.V. All rights reserved.

Digital holographic aberration compensation in electron holography

This paper proposes a new digital method to compensate for the aberration of an electron objective lens in electron holography. In this method, the object wavefront in the exit pupil plane is numerically reconstructed from a digitized electron hologram, and is corrected by multiplying it with the conjugated phase-error function. Then, an aberration-free image can be obtained by calculating the Fresnel integral of this corrected wavefront. In comparison with traditional methods, this method is much more convenient and accurate. Some verifying experiments are also presented in this paper. (C) 2003 Society of Photo-optical Instrumentation Engineers.