Criteria for correction of quadratic field-dependent aberrations

Aberrations of imaging systems can be described by using a polynomial expansion of the dependence on field position, or the off-axis distance of a point object. On-axis, or zero-order, aberrations can be calculated directly. It is well-known that aberrations with linear field dependence can be calculated and controlled by using the Abbe sine condition, which evaluates only on-axis behavior. We present a new set of relationships that fully describe the aberrations that depend on the second power of the field. A simple set of equations is derived by using Hamilton's characteristic functions and simplified by evaluating astigmatism in the pupil. The equations, which we call the pupil astigmatism criteria, use on-axis behavior to evaluate and control all aberrations with quadratic dependence on the field and arbitrary dependence on the pupil. These relations are explained and are validated by using several specific optical designs.     

Imaging characteristics of a wavefront coding system with off-axis aberrations

Imaging characteristics are analyzed for a wavefront coding system suffering from off-axis aberrations such as primary astigmatism and primary coma. Some analytical expressions for the optical transfer function are obtained by using the stationary-phase method. These expressions give the relationship between the optical transfer function and the off-axis aberrations. Some cases are computed and illustrated graphically. It is shown that the wavefront coding system has a high tolerance to primary astigmatism and a low sensitivity to primary coma.     

Bispectral Imaging Through Unknown Deterministic Aberrations

Abstract

When the deterministic aberrations are known in an optical system, traditional de-blurring methods are effective. However, when the aberrations are difficult to quantify, such as telescope aberrations or the aberrations in the human eye, other methods are needed. One potential method for de-blurring an image that is formed from a system with unknown aberrations is the bispectral imaging method. It is a promising way to remove the effects of deterministic aberrations when random aberrations are present or artificially introduced into the system. Through computer simulations, we have found the optimal amount of random aberrations to have present in a system containing deterministic aberrations. This amount optimizes the image quality of the reconstruction at high light levels using 100 statistically independent aberrated images of the object. Defocus and several third-order aberrations were considered in the isoplanatic case. The performance of this method was characterized by reconstructing a point source and computing its Strehl ratio. These results are currently being used to incorporate the bispectral imaging method as part of a non-invasive technique to reconstruct high-resolution images of the back of the eye in human subjects.     

Polychromatic Off-axis Optical Image

The colour specification in the polychromatic point-spread function leads to the integration for each primary colour of the monochromatic point-spread functions, taking account of the spectral emission of the source and the spectral response of the receiver. A numerical technique for the off-axis object point has been programmed. The colour variation on the off-axis image in the presence of the primary aberrations is discussed in terms of the illuminance and the chromaticity, and it is compared with that of the aberration free system. The study has been done for the standard observer (CIE 1931). Different correction types of transverse chromatic aberration, the most influential one on the chromaticity variation of the image, are studied, and the relation between the shape of this residual aberration curve and the quality of the image with respect to the illuminance and the chromaticity are shown. Simultaneously the influence of the spectral distribution of the source is shown.     

Aberrated Optical Systems with Optimum Apodizers for the Strehl Ratio: Axial and Extra-axial Point-spread Function and Modulation-transfer Function

Abstract

The amplitude distribution at the aperture of an optical system for off-axis object points due to the introduction of a non-uniform transmission filter depends on the transmission function and on the longitudinal filter position. The effects of several filters on the axial and extra-axial point-spread function (PSF) and modulation-transfer function (MTF) of an optical system for different filter positions are studied. The filter transmission function and the filter position are obtained by the optimization of the axial and extra-axial Strehl ratios. The axial and extra-axial effects of the filters on the PSF and on the MTF are analogous. There are some positions of the filter for which the overall performance is significantly improved. The optimum longitudinal positioning of the apodizing filter is achieved in a process which is analogous to the stop shift effect in conventional optical design. The lack of radial symmetry in the off-axis wavefront aberration function at any particular field angle is convolved with the apodizing function and this leads to the anticipated improvement.     

Broadband FUV imaging spectrometer: advanced design with a single toroidal uniform-line-space grating

Performances of a far-ultraviolet (FUV) imaging spectrometer in an advanced design are presented with a toroidal uniform-line-space (TULS) grating. It provides high spatial resolution and spectral resolution for a broadband and a wide field of view. A particular analysis for the grating aberrations, including all the high-order coefficients neglected by previous existing designs, was generated for indicating their significance. The analysis indicates that these high-order off-axis aberrations would have a remarkable influence on the design results. The transcendental equations composed of these aberration coefficients do not have analytic solutions in algebra. To solve the problem, the past designs always do some simplified calculation which only suits a narrow field of view and waveband. Thus, the optimization of the genetic algorithm is introduced to propose reasonable ranges of optical parameters. Then ZEMAX software is used to obtain the final optical system from these ranges. By comparing different design results of the same example, our advanced TULS design performs better than conventional TULS design and spherical varied-line-space grating design, and as well as the toroidal varied-line-space design. It is demonstrated that aberrations are minimized when the TULS design is operated by our method. The advanced design is low-cost, easy to fabricate, and more suitable for FUV observations.     

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.     

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.     

Transformation of Reference Spheres by an Aberration Free and Infinitely Large Optical System in the Fresnel Approximation

Imaging of a complex distribution by a thin lens is studied and generalized by analysing the imaging process of a complex distribution defined on an off-axis object sphere in an optical system. First, imaging of an object in an elementary system is considered, and then the considerations for the system being a combination of the elementary systems are presented. The general rule of reference spheres transformation is introduced. It is shown that complex distributions given on an off-axis object sphere are fully reconstructed on an image sphere of the system with a change of scale only. The relations between the parameters of the object and image spheres are determined. Some examples of the applications of the transforming rule are given.     

Three-dimensional display of a horizontal-parallax-only hologram

We propose a three-dimensional (3D) holographic display by converting an optically recorded complex full-parallax (FP) hologram to an off-axis horizontal-parallax-only (HPO) hologram. First, we record the complex FP hologram of an object using optical scanning holography. We then convert the complex FP hologram to an off-axis HPO hologram through fringe-matched Gaussian low-pass filtering and with the introduction of an off-axis reference. Finally, we reconstruct the off-axis HPO hologram optically using an amplitude-only spatial light modulator. Until now, only computer-generated HPO holograms have been displayed optically. To the best of our knowledge, this is the first demonstration of a 3D display of an optically recorded HPO hologram.     

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.     

New plane grating monochromator with off-axis parabolical mirrors

The design of a new plane grating monochromator is described. The system employs off-axis parabolical mirrors both in the collimator and the camera. The entrance and exit slits lie on the same side of the grating and are curved to obtain the invariance of the curvature of the spectral lines with respect to wavelength variations. The grating is oriented to avoid multiple dispersions. The system is free from aberrations in the center of the field, and since the dependence of the aberrations on any other point of the field is very weak, it is possible to use long slits. The images obtained with this design are of excellent quality and may be improved even further using mirrors with slightly different off-axis angles and focal lengths for the collimator and the camera. Analytical expressions of the aberrations are obtained by the plate diagram method; the results are checked by finite ray tracing for a particular case.     

Off-axis mirror based optical system design for circularization, collimation, and expansion of elliptical laser beams

In this paper, we present two optical system design methods for beam circularization, collimation, and expansion of semiconductor laser output beam for possible application in LIDAR systems. Two different optical mirror systems are investigated: an off-axis hyperbolic/parabolic mirror system and an off-axis parabolic mirror system. Equations specific to these mirror systems are derived and computer package programs such as ZEMAX and MATLAB are used to simulate the optical designs. The beam reshaping results are presented.     

Alternative uses of coddington's equations in optical design

Abstract

Considering a second-order patch surrounding an axial reference object point, formulas for the second field derivatives of the wavefront aberration function, corresponding to rays both in the tangential and sagittal sections, are given. To obtain these derivatives, Coddingtons equations are used in a way alternative to that employed to calculate the second aperture derivatives. The wavefront aberration function for any point in the patch is written in terms of data acquired tracing tangential rays from the axial point alone. The effectiveness of the procedures is tested numerically in two photographic objectives. The plots for the field derivatives can be incorporated to the traditional ones to improve the global optimization of the optical system.     

Ray-tracing and Aberration Formulae for a General Optical System

Abstract

All aspects of the ray-tracing and of the calculation of the monochromatic and chromatic aberrations for a completely general optical system are treated. These include the specification of the system, the ray-tracing formulae for refraction, reflection and transfer, the introduction of pseudo-paraxial variables in the object and image spaces, the determination of the pupil domain and the best image plane, plus calculation of aberrations along rays. Hopkins' canonical coordinates are employed in the object and image space to facilitate the calculation of the aberration polynomial coefficients, for use in the evaluation of diffraction-based image criteria, such as the point spread function. All the formulae involved are always determinate and of good accuracy, no matter whether the object, image or either pupil, of the system is at finite distance or at infinity.     

Two-beam interferometer for measuring aberrations of optical components with axial symmetry

We present a concept of interferometric testing, believed to be novel, that can be applied to measuring aberrations of optical components that have rotational symmetry. The optical configuration uses two coherent, collimated wave fronts that are tilted to impinge upon the optical component being tested such that one beam is on axis and the other is off axis. For small tilt angles the two aberrated wave fronts can be considered to be carrying the same aberrations. Furthermore, the off-axis beam is displaced along a direction orthogonal to the optical axis of the component. Interference between the two aberrated wave fronts produces a fringe pattern that is similar to a lateral shear interference pattern. Moiré fringes are obtained by spatial beating of the interference pattern with a CCD TV camera array. Under such conditions it is possible to subtract most of the linear carrier that is intrinsically present in the resultant fringe pattern owing to the large defocus aberration and tilt.     

Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.     

Exact ray-trace beam for an off-axis paraboloid surface

When an off-axis paraboloidal mirror focuses a parallel beam, the image is formed on one side of the optical axis. For a tilted beam focused by an off-axis paraboloidal mirror, the focus is no longer pointlike (not considering the diffraction effect); rather, it is a distorted spot. This is due to the inherent aberrations of the surface. In addition, there is a change in the focus position. We calculate by exact ray-trace equations the modified wave-front aberration and express it in power series. Our formulation uses the optical path variation along a defined principal ray that we relate to the parameter that describe the surface and the beam angle of incidence. We designate this ray as that reflected by the center of the entrance pupil and field of view. We employ the direction cosines of the principal ray to compute the wave-front aberration function of a beam reflected by an off-axis paraboloid.     

Active aberration correction for the writing of three-dimensional optical memory devices

We describe an active optical system that both measures and corrects the aberrations introduced when writing three-dimensional bit-oriented optical memory by a two-photon absorption process. The system uses a ferroelectric liquid-crystal spatial light modulator (FLCSLM) configured as an arbitrary wave-front generator that is reconfigurable at speeds as great as 2.5 kHz. A method of aberration measurement by the FLCSLM wave-front generator is described. The same device is also used to correct the induced aberrations by preshaping the wave fronts with the conjugate phase aberration as well as to scan the focal spot in three dimensions. Experimental results show the correction of both on- and off-axis aberrations, allowing the writing of data at depths as great as 1 mm inside a LiNbO3 crystal.