Third-order Aberration Coefficients Research Articles

HOE recording with non-spherical waves

A general study of third-order aberrations of aspheric holographic optical elements is described, including the effect of wavefront deviation from construction waves on the total aberration. Third-order aberration coefficients for a plane parallel glass plate referred to a general case in which the optical axes are not normal to the plate surfaces are calculated. This theoretical development is used to optimize the glass plate substrate aberration contribution in a holographic compound system used for optical image formation.

A study of first- and third-order chromatic aberrations for electrostatic round lenses

In this paper the first- and third-order chromatic aberrations of electrostatic round lenses have been studied based on the electron optical-canonical aberration theory. The present work uses the Hamiltonian function expansion method to derive all analytical expressions of the first- and third-order chromatic aberration coefficients instead of the generalized integration transformation method. Its advantage is that it is much simpler. An application to bi-potential immersion lens has been illustrated. It is hoped that this work would be helpful for investigating high-order electron optical aberrations in rotationally symmetric lenses, including geometric aberrations.

Optical properties of mirrors for focusing of non-normal incidence atom beams

Calculations of focal properties and third-order aberration coefficients are presented for atom mirrors based on elastically deformed single-crystal surfaces. Biaxial loading of the mirrors is achieved by a combination of a uniform applied pressure and rectangular boundary conditions. The results are calculated for readily available single-crystal wafers of Si(111) with a thickness of 50 μm and a usable free standing diameter of 18 mm. Focusing of beams at non-normal incidence requires the principal radii of curvature, in planes normal to the mirror plane, to be different. The ratio of these principal radii of curvature is shown to be insensitive to the applied pressure and can be varied by a factor greater than 10 by changing the boundary conditions from square (1:1) to rectangular (1:2).

Distortion aberration in a symmetric magnetic doublet for an electron beam projection system

It is found that large radial and azimuthal distortions are generated if the geometric conditions defined for the symmetric magnetic doublet lens system are not met for the inner and outer radii of the lens casing. These distortions are found by trajectory calculation and by computing third-order aberration coefficients. When the inner radius of the object side lens casing is decreased, then pincushion distortion is generated. When the outer radius of the cylindrical object side lens casing is decreased, barrel distortion is generated. Anisotropic distortion is generally present, changing sign when the inner radius of the casing is decreased significantly. The radial distortion can be compensated by adjusting the inner and outer radii of the lens casing, whereas the azimuthal distortion cannot be compensated by such adjustment. The origin of these distortions can be well understood by studying the magnitude of the maxima and minima of the axial magnetic field and magnetic vector potential. Field curvature, astigmatism, coma radius, coma length, axial chromatic, and spherical aberrations are not influenced by such variations of the inner or outer radii of the lens casing.

A transmission x-ray microscope based on secondary-electron imaging

A design for a transmission x-ray microscope with 20 nm transverse spatial resolution is presented. The microscope, which is based on the electron-optical imaging of the photoemitted electrons from an x-ray shadowgraph, consists of a transmission x-ray photocathode coupled to a photoelectron emission microscope (PEEM—also called a PEM for photoelectron microscope). Unlike the conventional PEEM, which produces a surface map of photoelectron yield, this microscope can provide information on the subsurface properties of thin samples. The analysis of the microscope’s electron-optical performance is based on the evaluation of Gaussian focusing properties and third-order aberration coefficients computed using several complementary methods. The electron optical properties of the microscope are examined with an emphasis on issues affecting overall performance and achieving the best possible resolution. Preliminary experimental results using a cesium iodide photocathode are shown.

Optical properties of planar immersion objectives in electron impact ion sources

The ion optical properties of an immersion ion source have been calculated in the presence of electron space charge. The ion source was an electron impact ionizer with planar geometry and a crossed beam configuration, assuming a well defined atomic beam. We present the focal properties and the third-order aberration coefficients, as a function of both the electron beam current and the ion extraction geometry. The image position is relatively unchanged by the electron beam or extraction geometry; however, there are significant changes to the magnification. The aberrations indicate that it is possible to preserve a spatial resolution in a typical imaging ion source of over a 6 mm field of vision. .

Third-order ion trajectory calculations of Wien filters

Three-dimensional ion trajectories in Wien filters were calculated in a third-order approximation. The results were expressed as a total transfer matrix with aberration coefficients. Two-dimensional image shapes at an image plane were simulated by using these results. Focusing properties of the systems with a homogeneous Wien filter and a stigmatic Wien filter were investigated as examples. Third-order aberration coefficients of the system were calculated and two-dimensional image shapes were pictorially estimated. The method for evaluating the ion optical properties of Wien filter systems was developed.

Four-mirror imaging system (magnification +1/5) for ArF excimer laser lithography

Research was conducted on the axially symmetric four-mirror system (magnification +1/5) for use in optical lithography using an ArF excimer laser beam. The initial design is derived from an extensive numerical calculation that makes the sum of the third-order aberration coefficients very small ∑j=IV|Sj|<10-6. Using an optimization method (damped least-squares method) the finite aberrations are reduced; then, to obtain the diffraction-limited performance, three surfaces are aspherized. The final system has NA=0.38 for the ArF excimer laser line (λ=0.193 μm) and depth of focus of 1.2 μm over a 2.6×2.6 cm2 object field. Nearly all rays fall within the Airy disk in the image plane. The resolution is 540 cycles mm-1 at MTF=0.5 level for axial object point. We consider that the present four-mirror system may be further refined for use in soft x-ray lithography.

Symmetric Magnetic Doublet Optics with Dynamically Compensated Field Aberration for Reducing Image Projection System

Aberrations calculated from the electron trajectories are compared with those estimated from the third-order aberration coefficients. Field curvature and radial and azimuthal distortions are consistent with each other for moderate radial value R. In order to optimize lens positions, a method is adopted which makes the lens condition under which the principal ray trajectory from the largest objective radius R o crosses the optical axis at its normal position equal to that under which the trajectories diverging from the same Ro focus on the image. For the 50 cm object-to-image distance optics, residual aberrations are calculated through the trajectory calculation after the corrections of lens current, crossover and beam position. When the main and subfield sizes are 20 mm and 0.25 mm square, the residual field curvature, astigmatism, and radial and azimuthal distortions are 13.1 nm, 6.4 nm, 19 nm, and 8.35 nm, respectively, when the beam semiangle is 0.1 mrad, and the beam energy is 100 keV.

Chromatic aberration of three-cylinder electrostatic lenses

Accurate calculations of the axial chromatic aberration coefficients of geometrically symmetric three-cylinder tripotential electrostatic lenses are presented for two different center electrode lengths. This is an extension of the first-order properties and the third-order spherical aberration coefficients published by Harting and Read.

Aberrations of holographic lenses in image quality evaluation

The imaging quality of holographic lenses depends on parameters that include the shape of a holographic lens surface or an input pupil position. Based on the formulas for third-order aberration coefficients derived for such cases, conditions that ensure the correction of aperture and field aberrations are given. The possibility of joint correction of spherical aberration, coma, and astigmatism is discussed. The formulas presented are illustrated with a number of examples; two types of holo-lenses are taken into account: <i>imaging</i> and <i>focusing</i>. For imaging quality assessment an aberration spot calculation method based on numerical evaluation of an appropriate diffraction integral is used. The results of this method are compared with the results of imaging quality estimation using the geometrical ray tracing method.

Lens first-order transfer matrices incorporating third-order aberrations

Abstract Transfer matrices of electrostatic or other lenses are useful for ray-tracing through combinations of two or more lenses [see e.g. J.H. Moore et al., Building Scientific Apparatus (Addison-Wesley, 1983)]. In this letter we present a simple modification than can be made to the first-order transfer matrix of a lens to incorporate the effects of all meridional third-order aberrations. The modified transfer matrix is intended for use within computer programs in which the third-order spherical aberration coefficients C S for point axial objects are either stored or can be calculated using a suitable numerical approximation [such as that given by E. Harting and F.H. Read, Electrostatic Lenses (Elsevier, 1976)].

Third-order aberration theory for weak gradient-index lenses

A set of simple analytic expressions has been developed for calculating the third-order aberration coefficients in weak or nearly thin axial and radial gradient-index lenses.

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.

Chromatic aberration of symmetric two-cylinder electrostatic lenses

Accurate calculations of the axial chromatic aberration coefficients of symmetric two-cylinder bipotential electrostatic lenses for three different gap values (0.10D, 0.50D, and 1.0D) are presented. This is an extension of the first-order properties and the third-order spherical aberration coefficients published by Harting and Read.

Image analysis of curved gradient-index rods

A method of ray tracing through curved gradient-index imaging rods is developed. A study of the imaging ability of a radial gradient-index relay rod, with end-to-end imaging of an object in the meridional plane, bent along some simple curves is performed. The study includes an analysis of changes in the first order of properties and the transverse aberrations in the image plane. Approximations of the first-, second-, and third-order transverse aberration coefficients are made.

Aberrations in curved graded-index fibers

Graded-index fibers can be efficiently used as image relays and therefore it is necessary to study the aberrations in the images transmitted through these media. Since curvature in fibers is inevitable, aberrations in curved graded-index fibers are also important. In this paper, we have obtained explicit expressions for the second- and third-order aberration coefficients in curved fibers which reduce to the ones for the straight fiber as the radius of curvature of the fiber tends to infinity. It is shown by numerical calculations that the total aberrations (up to third order) in a graded-index fiber depend critically on the values of grading parameters α and β. It is also shown that for small values of grading parameter, α ∼ 2 cm−1, curvature effects are significant for radii of curvatures less than ∼ 25 cm, whereas for larger values of the grading parameter, α ∼ 50 cm−1, curvature affects the aberrations for radii of curvatures less than about 10 cm.

Systematic transformations of the asymptotic aberration coefficients of round electrostatic lenses (1)

In previous work we formulated the third-order asymptotic aberration coefficients of round (axially symmetric) electrostatic lenses in a form independent of object and aperture positions, and expressions for the six quantities which are sufficient to specify completely the aberration properties of the lenses were derived in the form of integrals involving derivatives of the axial potential through the fourth order. Because actual calculations involved numerical differentiation of the axial potentials, integrations by parts were used to transform the integrals to two new forms with axial derivatives of lower degree. Many other forms of the aberration integrals can be obtained by further integrations by parts, but the transformations are laborious and it is not easy to predict the forms which are possible nor to determine the sequence of operations which will yield a desired result. However, using a method originally developed by Seman and extended by Hawkes, a completely general formula has been derived from which all of the possible forms of the asymptotic integrals can be obtained simply. A few of these possible forms are derived and discussed.

Design of single element gradient-index collimator

Gradient-index materials will provide new valuable degrees of freedom for the lens designer. The gradient-index single element lens is studied in terms of its third-order aberration coefficients, ray-trace analysis, and wave-front error.

Third Order Aberration of Conical SELFOC Fibers+

In this paper, we have obtained explicit expressions for the third-order aberration coefficients of conical SELFOC fibers, in which, in addition to a transverse variation of refractive index, there exists a dependence of refractive index along the axis of symmetry. We have also obtained the variation of the aberration coefficients with various parameters and have shown that there exists realistic values of the parameters for which spherical aberration, coma or distortion can be made zero. The analysis should find application in the design of gradient-index lenses.