Frequency response characteristics of a birefringent lens with off-axis aberrations

We report the frequency response characteristics of an optical system consisting of a lens made of a uniaxial birefringent crystal sandwiched between two linear polarizers; the lens has prespecified off-axis aberrations such as primary astigmatism and primary coma. An analytical expression is obtained for the optical transfer function of the proposed system by use of the autocorrelation of the pupil function over the lens aperture. Some specific cases are computed and illustrated graphically. It has been shown that the proposed system has imaging characteristics distinctly different from those of an ordinary glass lens, and these may be advantageous for better balancing of aberrations in conventional imaging systems.     

Diffraction patterns formed by an off-axis paraboloid surface

We derive a general, closed-form expression for the diffraction patterns including the aberrations that are due to off-axis alignment and positioning, of a paraboloid mirror. The diffraction patterns obtained in the focal plane of an off-axis paraboloidal mirror suffer modifications by the aberrations that are inherent in these surfaces: astigmatism and coma. Different magnifications in two perpendicular spatial directions indicate astigmatism. The Airy function, which affects a single spatial coordinate, describes the coma aberration. We identified the coma by the increased number of intensity zeros within adjacent lobules.     

Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism

A quartic phase retardation function is described that reduces the variation of the intensity of the focal point of incoherent imaging systems suffering from primary third-order aberrations limited to coma and astigmatism. Corresponding modulation transfer functions are shown to remain practically invariant for moderate amounts of coma and astigmatism.     

Geometric theory of wavefront aberrations in an off-axis spherical mirror

We present, analyze, and evaluate expressions for the wavefront aberrations in an off-axis spherical mirror. These formulas are derived from the optical path difference between an ellipsoid and a sphere, assuming a relatively small pupil and a small angle of incidence, as will be described in detail. Some well-known and also some useful new aberration expressions are obtained. They can be used to design and analyze cavities, spectrographs, and retinal adaptive optics imaging systems.     

Geometrical theory of aberrations near the axis in classical off-axis reflecting telescopes

A geometrical theory of aberrations for the vicinity of the focus of arbitrary off-axis sections of conic mirrors is derived. It is shown that an off-axis conic mirror introduces linear astigmatism in the image. However, in classical two-mirror telescopes this aberration can be eliminated by tilting the secondary parent mirror axis. It is also shown that the practical geometrical-optics performance of a classical off-axis two-mirror telescope with no linear astigmatism is equivalent to the performance of an on-axis system, proving that both systems have identical third-order coma. To demonstrate the applicability of the theory developed in a practical system, a fast (i.e., f/2), compact, obstruction-free classical off-axis Cassegrain telescope is designed.     

Techniques for measuring aberrations in lenses used in photolithography with printed patterns

In optical lithography, it is a serious problem that aberrations in projection lenses reduce the imaging quality. Therefore techniques to measure the aberrations are required that will predict the adverse effects of aberrations on lithographic imagery and reduce them. We present a measurement method that uses a fine grating and its imaging condition to quantify coma, astigmatism, and spherical aberration. With this method, these aberrations can be described with simple expressions from the measured results. Application of this method revealed the coma of Zernike polynomials for our krypton fluoride (KrF) excimer-laser scanner.     

Mosaic imaging with spatial light modulator technology

Diffractive wavefront control with spatial light modulator technology has been successfully demonstrated both as a means of steering optical wavefronts and for compensating large optical aberrations. We describe the use of a spatial light modulator operating as a programmable diffractive optic to record extended-field-of-view mosaic images centered about a large off-axis field angle. A gimbaled primary mirror allows the nominal viewing angle to be varied, and diffractive wavefront steering allows images to be acquired at discrete increments in the field angle about the nominal angle. Diffractive wavefront compensation corrects aberrations associated with the primary mirror orientation and changes in field angle introduced by diffractive wavefront steering. A time sequence of images recorded at discrete increments in the field angle is then digitally combined to create a high-pixel-count mosaic image. The effects of diffraction efficiency and wavelength-dependent wavefront errors on image quality are analyzed.     

The design of a triplet with a shape-dependent third-order aberration optimization technique

Abstract

We have derived a series of shape-dependent third-order aberration equations involving lens thickness. These equations are used in a simple and direct method to correct real aberrations and to find the minimum aberration for triple design that has been proposed. These calculated third-order aberrations can exactly meet the target values for each stage by means of the damped least-squares method. The shape of the three elements permits the control of three third-order aberrations: spherical aberration, coma and astigmatism. The spherical aberration is targeted first. The minimal value of the real on-axis aberration is obtained. Similarly, the coma and astigmatism target values are adjusted so as to force the full field angle real off-axis aberrations down to a minimum. Finally, the manual adjustment of the lens thickness and air spacing is used to attain the minimum aberration at the 0.7 field angle. To verify the method, two triplet design examples are presented.     

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.     

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%.     

Experimental study of an off-axis three mirror anastigmatic system with wavefront coding technology

Wavefront coding (WFC) is a kind of computational imaging technique that controls defocus and defocus related aberrations of optical systems by introducing a specially designed phase distribution to the pupil function. This technology has been applied in many imaging systems to improve performance and/or reduce cost. The application of WFC technology in an off-axis three mirror anastigmatic (TMA) system has been proposed, and the design and optimization of optics, the restoration of degraded images, and the manufacturing of wavefront coded elements have been researched in our previous work. In this paper, we describe the alignment, the imaging experiment, and the image restoration of the off-axis TMA system with WFC technology. The ideal wavefront map is set to be the system error of the interferometer to simplify the assembly, and the coefficients of certain Zernike polynomials are monitored to verify the result in the alignment process. A pinhole of 20 μm diameter and the third plate of WT1005-62 resolution patterns are selected as the targets in the imaging experiment. The comparison of the tail lengths of point spread functions is represented to show the invariance of the image quality in the extended depth of focus. The structure similarity is applied to estimate the relationship among the captured images with varying defocus. We conclude that the experiment results agree with the earlier theoretical analysis.     

Alteration of imaging properties of an optical pickup's objective lens by a nonrotationally symmetric coating

One can control the aberrations of an optical readout system by varying the width of a strip of antireflection coatings deposited upon plastic objective lenses. It is found that one can control the magnitude of the third-order astigmatism of the system by changing the coating width. This process has the advantage that it does not significantly cause other kinds of aberration such as coma and spherical aberrations to deteriorate. When these nonrotational symmetrically (NRS) coated lenses are used for off-axis operations such as tracking movements in optical drives, the change in the magnitude of the astigmatism (DAS) that is generated can be made smaller than those of symmetrically coated or noncoated lenses. As much as a 73% decrease in DAS was observed experimentally with a NRS-coated lens. Including the birefringence of the plastic material in the analysis yields a low and constant level of astigmatism generated by shifting of the objective lens.     

Aberration measurement from specific photolithographic images: a different approach

Techniques for measurement of higher-order aberrations of a projection optical system in photolithographic exposure tools have been established. Even-type and odd-type aberrations are independently obtained from printed grating patterns on a wafer by three-beam interference under highly coherent illumination. Even-type aberrations, i.e., spherical aberration and astigmatism, are derived from the best focus positions of vertical, horizontal, and oblique grating patterns by an optical microscope. Odd-type aberrations, i.e., coma and three-foil, are obtained by detection of relative shifts of a fine grating pattern to a large pattern by an overlay inspection tool. Quantitative diagnosis of lens aberrations with a krypton fluoride (KrF) excimer laser scanner is demonstrated.     

Optimization method for ultra-wide-angle and panoramic optical systems

To an ultra-wide-angle and panoramic optical system, the aberrations of point object at any field angle are separated into two types: the aperture-ray aberrations of off-axis point object and the chief-ray aberrations. A simple form of the triangular formulae of tracing an oblique-incidence ray is derived to calculate the chief-ray parameters and their aberrations; moreover, the aperture-ray aberrations of an off-axis point object are analyzed with the plane-symmetric aberration theory. Based on the two types of aberrations, we present a merit function for ultra-wide-angle and panoramic optical systems; the optimization program with the differential-evolution algorithm is then developed. To validate the optimization method we finally optimize a fish-eye lens and a catadioptric omnidirectional imaging system.     

Wavefront-guided correction of ocular aberrations: are phase plate and refractive surgery solutions equal?

Wavefront-guided laser eye surgery has been recently introduced and holds the promise of correcting not only defocus and astigmatism in patients but also higher-order aberrations. Research is just beginning on the implementation of wavefront-guided methods in optical solutions, such as phase-plate-based spectacles, as alternatives to surgery. We investigate the theoretical differences between the implementation of wavefront-guided surgical and phase plate corrections. The residual aberrations of 43 model eyes are calculated after simulated refractive surgery and also after a phase plate is placed in front of the untreated eye. In each case, the current wavefront-guided paradigm that applies a direct map of the ocular aberrations to the correction zone is used. The simulation results demonstrate that an ablation map that is a Zernike fit of a direct transform of the ocular wavefront phase error is not as efficient in correcting refractive errors of sphere, cylinder, spherical aberration, and coma as when the same Zernike coefficients are applied to a phase plate, with statistically significant improvements from 2% to 6%.     

空间光学合成孔径成像系统原理与关键技术分析

根据傅里叶成像理论,分析了光学合成孔径系统成像原理,指出光瞳函数的离散化,提高了光学合成孔径成像系统的分辨本领,但降低了系统传函中频性能,并使系统在信息获取时具有方向选择性。运用遗传算法对光瞳构形进行优化,光瞳优化后,四孔径系统的优化参数从310.8降到13.7,减小了23倍。在合成孔径成像系统单个子孔径存在像差的情况下,Piston误差对系统的影响最大,其次是离焦误差、倾斜误差、球差、彗差和像散,对于四孔径系统,Piston误差对系统的影响几乎是离焦误差的2倍,像散的4倍。系统曝光时间与填充比的平方或立方成反比。     

Nonaxial Strehl ratio of wavefront coding systems with a cubic phase mask

An approximate expression of the peak position of the point-spread function (PSF) of wavefront coding systems with a cubic phase mask (CPM) is derived and verified by simulation results. An approach called the nonaxial Strehl ratio (NASR) is used to evaluate the performance of wavefront coding systems with defocus aberrations. The characteristics of the NASR are investigated. The relationships between the NASR and the similarity of out-of-focus modulation transfer function (MTF) and recoverability of blurred encoding images are presented, and some useful guidelines are given for the optimization of phase mask parameters according to these relationships.     

Generalized optical ABCD theorem and its application to the diffraction integral calculation

We generalize the transfer matrix ABCD theorem for paraxial rays of the optical system to skew rays propagated off axis, whether or not the system possesses rotational symmetry. Furthermore, we apply the generalized ABCD theorem to evaluate the diffraction integral matrix elements A-D expressed in terms of the angle eikonal T, with the primary aberrations included. Finally, analysis and numerical calculation are given for propagation of a light beam through the optical system in the case in which spherical aberration and coma are present.     

Holographic Grating Aberration Correction for a Rowland Circle Mount II

We generalize the treatment given in Part I to show how to correct primary astigmatism or primary coma at two wavelengths in two orders. We then introduce a further state of correction that permits the simultaneous correction of primary astigmatism and primary coma for a given value of 1 ? m λ/λ⁰ ? 2. Some practical implications are discussed.     

Implementation of sine condition test to measure optical system misalignments

Rather than measuring aberrations across the field to quantify the alignment of an optical system, we show how a single, on-axis measurement of the pupil mapping can be used to measure the off-axis performance of the system and determine the state of alignment. In this paper we show how the Abbe sine condition can be used to relate the mapping between the entrance and exit pupils to image aberrations that have linear field dependence. This mapping error then can be used to measure the linear astigmatism caused by the misalignment. Additionally, we present experimental results from the sine condition test on a simple system.