Gaussian Image Plane Research Articles

It is a sufficient condition only, not a necessary and sufficient condition, for decomposing wavefront aberrations

The classic equation for decomposing the wavefront aberrations of axis-symmetrical optical systems has the form, $$ W({h}_0,\rho,\phi )=\sum_{j=0}^{\propto } \sum_{p=0}^{\propto } \sum_{m=0}^{\propto } {C}_{\left(2j+m\right)\left(2p+m\right)m}({h}_0{)}^{2j+m}(\rho {)}^{2p+m}(\mathrm{cos}\phi {)}^m $$ where j, p and m are non-negative integers, ρ and ϕ are the polar coordinates of the pupil, and h0 is the object height. However, one non-zero component of the aberrations (i.e., C133h0ρ3cos3ϕ) is missing from this equation when the image plane is not the Gaussian image plane. This implies that the equation is a sufficient condition only, rather than a necessary and sufficient condition, since it cannot guarantee that all of the components of the aberrations can be found. Accordingly, this paper presents a new method for determining all the components of aberrations of any order. The results show that three and six components of the secondary and tertiary aberrations, respectively, are missing in the existing literature.

Теория и расчет электростатических электронных зеркал с учетом релятивистских эффектов

Using the central particle method, the equations of the trajectory of charged particles in an axisymmetric electrostatic mirror are obtained with an accuracy of the third order of smallness inclusive with account for relativistic effects. The conditions for spatial focusing and the coefficients of spatial aberrations in the Gaussian plane of the mirror image are determined with account for relativism. By means of numerical calculations, conditions for simultaneous elimination of spherical and axial chromatic aberrations are determined, taking into account relativistic effects, in an axisymmetric electrostatic mirror when the object plane of the mirror is aligned with its focal plane. It is shown that account for high particle velocities leads both to a shift in the position of the Gaussian image plane and a change in the focusing quality.

Alternative Description for Gaussian Image Plane

An alternative description for the Gaussian image plane (GIP) of an optical system for a given object is presented, which applies to both aberration-free and non-aberration-free systems. We extend the definition of transverse magnification (TM) to the image plane (IP) displaced from the GIP and find that the TM depends linearly on the locations of both an aperture stop placed in front of the system and the IP. Hence, we redefine the GIP as the location at which the slope of the TM variance changes sign. The definition is deterministic and self-consistent and, therefore, no other parameters or measurements are needed. The derivation of this definition using a set of paraxial ray tracings and supporting experimental data for a thick bi-convex lens system is presented.

Characterization of the cathode objective lens by Real-Space Microspot Low Energy Electron Diffraction

Measurement of the geometric aberrations of the cathode objective lens is a key step in setting up a high quality C(c) and C₃ corrected state in state-of-the-art aberration corrected instruments. In addition to Cc and C₃, elimination of sample tilt and astigmatism is necessary for optimum imaging conditions. In transmission electron microscopy (TEM), the use of Zemlin tableaux has become the method of choice for measuring geometric aberrations, while Scanning TEM relies on the use of the ronchigram to assess the properties of the contrast transfer function. These techniques are not applicable to cathode lens instruments such as a low energy electron microscope. Instead, we propose to utilize Microspot Low Energy Electron Diffraction (LEED) with an illumination area on the order of 0.1 μm, where the image locations of the diffracted beams are observed in the Gaussian image plane. This Real-Space MicroLEED (RS-μLEED) pattern is the first derivative of the aberration function with respect to the angle of the diffracted beams. Thus, the geometric distortions observed in the RS-μLEED pattern are a direct measure of the relevant geometric aberrations.

Properties of the imaging performance of an electron optical system for SEM

The behavior of beam intensity distributions and optimum focus positions have been calculated as a function of beam convergence angle α at the image plane near the optimum focus condition for an optical system, including the effect of spherical aberration. In the diffraction-limited region, the beam intensity distribution can be modeled by a Gaussian and the image resolution is proportional to the FWHM of the beam intensity distribution. In this region, the FWHM of beam is proportional to 1/α and the optimum focus position measured from Gaussian image plane is proportional to α2, which is near the disk of least confusion due to spherical aberration. In the aberration-limited region, the FWHM of beam is independent of α and the beam tail radius increases with increase in α. In this region, the optimum focus is also independent of α; however it varies with the pixel size of the SEM image. The attainable resolution is obtained under the condition where the system makes a transition from diffraction-limited to aberration-limited, and at this point there is a minimum FWHM, a minimum tail of the beam and maximum axial intensity of the beam. The information passing capacity (IPC) of an optical system can be used to detect precisely the transition point.

Calculation of aberration coefficients by ray tracing

In this paper we present an approach for the calculation of aberration coefficients using accurate ray tracing. For a given optical system, intersections of a large number of trajectories with a given plane are computed. In the Gaussian image plane the imaging with the selected optical system can be described by paraxial and aberration coefficients (geometric and chromatic) that can be calculated by least-squares fitting of the analytical model on the computed trajectory positions. An advantage of such a way of computing the aberration coefficients is that, in comparison with the aberration integrals and the differential algebra method, it is relatively easy to use and its complexity stays almost constant with the growing complexity of the optical system. This paper shows a tested procedure for choosing proper initial conditions and computing the coefficients of the fifth-order geometrical and third-order, first-degree chromatic aberrations by ray tracing on an example of a weak electrostatic lens. The results are compared with the values for the same lens from a paper Liu [Ultramicroscopy 106 (2006) 220–232].

Aberration analysis of Cs‐corrector system with twin hexapoles and transfer lens doublet in scanning transmission electron microscope by simple ray tracing based on geometrical optics

AbstractOptical properties of scanning transmission electron microscopes with aberration correction systems, called Cs‐correctors, were simply calculated from the viewpoint of the geometrical optics. Here, we focused on the Cs‐corrector consisting of twin hexapoles and two round lenses. Although the above system has already been investigated theoretically and developed successfully, we attempted to interpret its properties by a simple ray tracing technique to understand them more intuitively. The aberrations on the Gaussian image plane of an objective lens were calculated. Analyses of the aberrations up to the fifth order indicated that hexapoles and round lenses could eliminate up to third‐order aberrations in the total optical system, including the objective lens, when these four elements are arranged properly. The fourth‐ and fifth‐order aberrations could also be suppressed by adding one and two round lenses, respectively. As these results agreed well with those of previous theoretical studies, our simple method is useful for designing an actual aberration correction system. Copyright © 2008 John Wiley & Sons, Ltd.

Calculation of Aberration Coefficients by Ray Tracing

In this paper we present an approach for the calculation of aberration coefficients using accurate ray tracing. For a given optical system, intersections of a large number of trajectories with a given plane are computed. In the Gaussian image plane the imaging with the selected optical system can be described by paraxial and aberration coefficients (geometric and chromatic) that can be calculated by least-squares fitting of the analytical model on the computed trajectory positions. An advantage of such a way of computing the aberration coefficients is that, in comparison with the aberration integrals and the differential algebra method, it is relatively easy to use and its complexity stays almost constant with the growing complexity of the optical system. This paper shows a tested procedure for choosing proper initial conditions and computing the coefficients of the fifth-order geometrical and third-order, first-degree chromatic aberrations by ray tracing on an example of a weak electrostatic lens. The results are compared with the values for the same lens from a paper Liu [Ultramicroscopy 106 (2006) 220-232].

Image assessment of a birefringent lens based on the factor of encircled energy

The factor of encircled energy (FEE) in the diffraction pattern formed by an optical system consisting of an aberration-free birefringent lens sandwiched between two linear polarizers is computed directly from the optical transfer function (OTF) of the system. FEE curves are given showing the variation of image quality of the proposed system at the Gaussian image plane and other observation planes with the birefringent lens parameters and the azimuthal settings of the transmission axes of the polarizers. The performance is compared to that of an ideal conven- tional glass lens. © 2003 Society of Photo-Optical Instrumentation Engineers.

The factor of encircled energy from the optical transfer function

This paper proposes a generalized technique for computing the factor of encircled energy (FEE) in a diffraction image from the optical transfer function (OTF) of the imaging system. This technique is applicable to diffraction-limited optical systems as well as to systems suffering from any aberrations, provided the transfer function of the same is known. It is also applicable to any kind of rotationally symmetric or asymmetric apertures. The accuracy of the results depends on the precision with which the OTF of the system can be determined. The FEE values are obtained using this technique for an aberration-free spherical lens as well as for the same having primary spherical aberration. The results are computed for the Gaussian image plane and other observation planes and are compared with the previously reported results. A numerical technique for computing the FEE in a diffraction pattern formed by a conventional lens suffering from off-axis aberrations and that formed by rotationally asymmetric apertures is also depicted.

Frequency response characteristics of a birefringent lens

The diffraction-based optical frequency response characteristic of a birefringent crystal lens sandwiched between two linear polarizers has been studied analytically using the autocorrelation of the pupil function over the lens aperture. The optical transfer function (oTf) of this proposed system depends on the material as well as design parameters of the crystal lens, along with the orientations of the transmission axes of the two polarizers relative to each other and to the optic axis of the birefringent lens. Naturally, the OTF of such a system can be varied at the lens fabrication stage like other lenses. An online variation of the OTF is also possible just by rotating any of the two polarizers included in the system. It has been shown that the proposed system can be adapted either for apodization or for super-resolution simply by rotating any of the two polarizers. This system may be designed for obtaining high depth of focus without significant image degradation, as well as for image processing operations like high frequency enhancement. The response of the system is studied at the Gaussian image plane as a function of the line frequencies in the object for different values of the birefringent lens parameters, and also for two different settings of the analyzer. The results are compared with the response of an ideal lens.

Parallel beams from nuclear microprobe lens systems

A nuclear microprobe lens system is normally used to focus a diverging beam into a fine probe. It does this by making the astigmatism terms in the first-order transfer matrix: ( x/ θ) and ( y/ φ) zero. Thus the position of a ray vector in the image plane ( x, y) in the linear approximation will not depend on the divergence of the ray vector in the object plane ( θ, φ). This is accomplished at the expense of magnifying the object divergence by factors ( θ/ θ) and ( φ/ φ) in the xoz and yoz planes, respectively, leading to relatively steep convergence angles at the Gaussian image plane where the specimen is located. An alternative operation mode of the nuclear microprobe lens system is possible where the convergence angle of the beam after the lens system in the linear approximation is made independent of the divergence angle of the beam prior to the lens system. In this case the terms ( θ/ θ) and ( φ/ φ) in the first-order transfer matrix are zero. This mode of the lens system does not produce a focused beam, so is most suitable for crystals without lateral structure. As an example, we present measurements from the 〈0 0 0 1〉 α-Al 2 O 3 axis, which show significant improvement in channeling angular widths for planar channeling mode even when compared to the unfocused mode.

Interference experiments with energy filtered electrons

Interference of purely inelastically scattered electrons in the Gaussian image plane of an energy-filtering TEM (EFTEM) provides a means to measure the localisation of a specific inelastic scattering process. Although, the initial degree of angular coherence given by the illumination angle of the electron source is reduced by the inelastic scattering event, interference of single plasmon scattered electrons was observed in a Zeiss EM 912 OMEGA. The lower limit of the localisation of single plasmon scattering in an aluminium single crystal was measured in this experiment to be 16 nm. Present experimental restrictions are the limited degree of coherence of the LaB 6 electron source and the fixed position of the Möllenstedt-biprism along the beam of the Zeiss EM 912 OMEGA electron microscope.

Performance of an Off-axis Holographic Lens in the Presence of Aberrations Induced by Construction-reconstruction Wavelength Shift: Extended Object Imaging

In this paper investigations have been carried out on the imaging of extended objects by an off-axis holographic lens operating at a wavelength different from its recording. Aberration balancing and correction techniques have been applied to improve the quality of imaging coherent as well as incoherent bar and edge objects. The light intensity distribution in the Gaussian image plane has been computed numerically and the results have been presented in graphical form. The image quality of the aberration-balanced holo-lens has been found to be comparable to that of the ideal holo-lens.

Imaging of Extended Objects Using an Off-axis Holographic Lens

Abstract The aim of this paper is to evaluate the performance of an off-axis holographic lens afflicted with third-order spherical aberration and coma using a numerical ray-tracing procedure. The light intensity distribution in the aberration spot observed at the Gaussian image plane has been evaluated for different amounts of coma and for a fixed value of spherical aberration. The images of both an extended coherent and an incoherent edge as well as bar objects have been computed and the results compared with the aberration-free case. All the results are illustrated in graphical form.

Imaging of Extended Objects Using a Curved Holographic Lens Free From Spherical Aberration

Abstract This paper evaluates the performance of an off-axis curved holographic lens using a numerical ray-tracing procedure. The hologram curvature has been suitably selected to eliminate spherical aberration. Light intensity distribution in the aberration spot observed at the Gaussian image plane has been evaluated for different object fields. The images of extended coherent and incoherent edge as well as bar objects have been computed and the results compared with that for the aberration-free case. All the results are illustrated in graphical form.

On the Inversion of an Integral Equation Relating Two Wavefunctions in Planes of an Optical System Suffering from an Arbitrary Number of Aberrations

If the wavefunction in the (not necessarily gaussian) image plane of an optical instrument is distorted by an arbitrary number of aberrations, the wavefunction in planes situated between the image plane and the plane of the specimen holder cannot be reconstructed by a Fourier series or a Fourier integral. This paper shows that the wavefunctions in those planes are uniquely determined by the values of the wavefunction in the image plane, and that an inversion formula, which in its structure is very similar to the well-known Fourier series, can be derived. Mathematically the problem concerns the inversion of the integral equation h (x 1, x 2) = Z † exp iS (x 1, x 2, y 1, y 2) ‚ (y 1, y 1) dy 1 dy 2 if the eikonal S is a multinomial.

X-Ray Imaging with Toroidal Mirror

X-ray imaging is made with a single toroidal mirror or two successive toroidal mirrors. Geometrical images at the Gaussian image plane are described by the ray trace. Application of a single toroidal mirror to small-angle scattering is presented.

Analysis of parallel-plate retarder in electrostatic zoom lens

A simple preretarding lens system is analyzed which involves a parallel-plate retarder B placed between two cylindrical lens systems A and C. The exact transfer matrix elements for the retarder are developed. The system can deliver fixed-energy electrons for any input energy by providing potentials on the upstream lens element which are a fixed fraction of the desired input kinetic energy. The location of the Gaussian image plane of the system is independent of the retarding factor for mode β and nearly so for mode α. In one mode, α, the retarder is at a crossover between lenses A and C. In the other mode, β, the retarder is anywhere in the field-free region between lenses A and C, and the rays from a given point in the object are parallel to each other near B. It is shown that the magnification in mode α can be independent of the retarding factor while that in mode β can be proportional to the square root of the retarding factor. The lens effect of the holes in the two grids is analyzed and a numerical example is given. Electron trajectories are shown for a particular example of lens systems A and C.

Velocity Measurement of Diffuse Objects by Using Speckle Dynamics

A new method of velocity measurement using dynamic properties of speckles produced at the gaussian image plane of a moving diffuse object is discussed theoretically and experimentally. The relation between the principles of this method and a laser Doppler velocimeter is made clear. It is shown that these two methods utilize different kinds of light intensity fluctuations and have complementary conditions for effective measurement. Experimental verification and application of the present method for the visualization of velocity distribution are also given.