Aberration Coefficients Research Articles

Relationship Between Aberration Coefficients of an Optical Device and Its Focusing Property

The best focus point of a focused Gaussian beam subject to a phase aberration is generally shifted with respect to the focal plane of the focusing lens. This focus shift is attributed to a lensing effect that belongs to the phase aberration, which mean focal length can be determined from the aberration coefficients determined in the framework of a Zernike polynomial decomposition. In this paper, we have checked the validity of this procedure, already available in literature, applied to three aberration types: a pure primary spherical aberration, the Kerr effect induced by a Gaussian beam, and an axicon illuminated by a Gaussian beam. Note that usually, the mean focal length of an aberrated lens is based on the relation between the effective radius of curvature of the wavefront before and after the lens. However, in this paper, the focal length associated with the phase aberration under study is defined from the point of the best focus, where the diffracted intensity on the axis is the maximum.

Linearized wavefront sensing model for aberration retrieval from low-frequency Fourier coefficients

This paper proposes and demonstrates a linearized model for phase diversity wavefront sensing, facilitating real-time processing and much less data required for training. Specifically, we find that the low-frequency Fourier coefficients of point spread function images are linearly proportional to pupil aberration coefficients under certain conditions. Simulation and experimental results show that the model can greatly reduce the processing time to several milliseconds by merely requiring hundreds of training samples while maintaining a comparatively high accuracy with state-of-the-art methods.

Design method of zoom systems based on high-order structural aberration coefficients

Design method of zoom systems based on high-order structural aberration coefficients

Design of electrostatic lenses through genetic algorithm and particle swarm optimisation methods integrated with differential algebra

Genetic algorithm (GA) and particle swarm optimisation (PSO) techniques have been integrated with the differential algebra (DA) method in charged particle optics to optimise an Einzel lens. The DA method is a robust and efficient tool for the calculation of aberration coefficients of electrostatic lenses, which makes use of nonstandard analysis for ray tracing a particle as it is subjected to the field generated by a lens. In this study, initial populations of lenses with random geometrical configurations are generated. These initial populations are then subjected to GA and PSO algorithms to alter the geometry of each lens for a set number of iterations. The lens performance is evaluated by calculating the spot size using the aberrations coefficients up to third-order generated by the DA method. Moreover, a focusing column comprising two lenses and a Wien filter was optimised using GA method.

Q-value-guided femtosecond laser-assisted in situ keratomileusis for low-to-moderate myopia with age-related accommodation deficiency

Objective: To evaluate clinical outcomes and visual quality after femtosecond laser-assisted in situ keratomileusis (FS-LASIK) performed with the Q-value-guided optimized monocular vision protocol (Custom-Q) for correction of low-to-moderate myopia and compensation for age-related accommodation deficiency. Methods: A prospective study was performed based on patients with myopia and age-related accommodation deficiency, who underwent Custom-Q FS-LASIK in the Peking University Third Hospital from October 2022 to October 2023. Monocular and binocular distance, intermediate, and near visual acuities, simulated whole-course visual acuity (binocular defocus curve), objective and subjective refractions, anterior corneal Q factor, anterior corneal higher-order aberrations and a subjective questionnaire assessing near visual acuity and visual quality were evaluated at 3 months postoperatively. Visual acuity was recorded using the logarithm of the minimum angle of resolution. The data were analyzed using the t-test and the Wilcoxon rank-sum test. Results: A total of 45 patients (90 eyes) who met the inclusion criteria and completed the 3-month follow-up were included in the study. The mean age of the 21 men (42 eyes) and 24 women (48 eyes) was (42.82±2.72) years. During the 3-month follow-up, all the patients achieved binocular uncorrected distance visual acuity not less than 0.00 and uncorrected near visual acuity not less than 0.20. Defocus curves revealed better results at 3 months postoperatively [0.00 (-0.08, 0.00), -0.08 (-0.08, 0.00)] at intermediate and near vergence (-1.50 to -1.00 D) compared to the preoperative values [0.00 (-0.08, 0.05), -0.08 (-0.08, 0.00)] (P<0.05). All the 45 patients maintained 0.10 or better vision at the defocus range from -2.50 to 0.00 D. The Q value in the dominant eyes was more positive postoperatively (0.01±0.26 vs.-0.15±0.14, P<0.05), while in the nondominant eyes, the Q value and corneal spherical aberration coefficient became more negative than those before surgery (-0.27±0.32 vs.-0.14±0.11, P<0.05; 0.08±0.05 vs. 0.12±0.03, P<0.05). The questionnaire demonstrated 42 (93.3%) patients were satisfied with near vision. The three most commonly reported visual discomforts after surgery were blurred vision, glare and halos. These symptoms were all mild to moderate in severity and did not interfere with daily life. Conclusion: For myopic patients with age-related accommodation deficiency, the Custom-Q algorithm proved to be a safe and effective way to achieve acceptable near vision and visual quality without compromising distance vision.

Analysis of the zoom system of a microscope condenser and its modification using variable focus lenses

A detailed theoretical analysis and optimization of the classical three-element zoom (pancratic) microscope condenser according to a patent from the 1930s [Reichspatentamt Nr. 713188 (29 10 1936)] is performed and formulas are derived for calculating basic parameters and the displacement of lenses during zooming. Furthermore, the modification of the classical zoom microscope condenser is investigated using a simpler optical system of two lenses with variable focal lengths and fixed positions. The relations for the calculation of the focal lengths of both variable focus lenses and the basic parameters of the zoom system have been described. The proposed two-element zoom system consisting of a system of two lenses with variable focal lengths maintains a constant distance between the object and image planes and fixed position of both lenses during zooming. The basic parameters and third-order aberration coefficients of such a system are calculated using an example.

Dynamics of charged particles in a solenoid and the accurate determination of the magnetic axis of a solenoid using measured field data

A fundamental understanding of the theory and measurement procedures for characterizing a solenoid is crucial for making a project cost-effective and achieving the desired performance for its use in a LINAC. This paper attempts to comprehend the beam dynamics of charged particles by considering the conservation of canonical angular momentum, which can take on positive, negative, and zero values. The analytical expression to generate a parallel beam using a solenoid is derived. The effect of the final beam size on the values of a solenoid’s spherical aberration coefficient and the position of the minimum beam size with respect to the focal length is studied using electron trajectories. A practical procedure to tune the above two parameters of a solenoid is demonstrated by simply modifying the demountable disks connected at each end. A mathematical model is proposed to accurately determine the axis of a solenoid and the magnetic centre using the measured magnetic field data, which is validated experimentally. Several solenoids, used in the 9.5 MeV 10 kW electron LINAC suitable for sterilizing medical products and food irradiation, have been characterised using this procedure.

Determination of the point resolution of high-resolution transmission electron microscope using the through-focus technique

From the viewpoint of evaluating the instrumental performance of high-resolution electron microscopy (HREM), the Scherzer condition was investigated using information theory. As a result, the optimum defocus amount Δf can be expressed based on (Csλ)12, and the formula Δf=1.12(Csλ)12 is obtained. Furthermore, a procedure for measuring point resolution using the through-focus technique is developed, and a new method for determining the spherical aberration coefficient using the variance of Δf is introduced in the procedure.

Optimization of optical system for reducing tolerance sensitivity based on the high-order nodal aberration theory

This paper proposes an optimization method aimed at addressing the challenge of balancing system imaging performance and tolerance performance. The method is based on the high-order nodal aberration theory. The calculation method for high-order aberration coefficients is derived, along with the change in aberrations after introducing tolerance perturbation. The evaluation system includes both the imaging performance of the system and its tolerance factors. A comprehensive merit function is created for optimization. Three aspherical system examples were designed for verification, and the optimization results of this method show significantly lower tolerance sensitivity compared to the general optimization results and the SAB function optimization results of CODE V.

Visual Quality Assessment After FS-LASIK Using Customized Aspheric Ablation Profile for Age-Related Accommodation Deficiency Compensation.

To evaluate clinical outcomes and visual quality 12 months after femtosecond laser-assisted laser in situ keratomileusis (FS-LASIK) performed with the Custom-Q algorithm for correction of myopia with or without astigmatism and compensate for age-related accommodation deficiency. Patients who had Custom-Q FS-LASIK for myopia and myopic astigmatism with age-related accommodation deficiency were included in this retrospective study. Distance, intermediate, and near visual acuities, objective and subjective refractions, Q-factor, corneal higher order aberrations (HOAs), accommodation function, defocus curve, contrast sensitivity, and a subjective questionnaire assessing visual quality were evaluated 12 months postoperatively. Clinical data of 43 cases were analyzed. The mean age was 42.02 ± 1.85 years (range: 40 to 48 years). At the 12-month follow-up visit, there were 43 (100%), 42 (97%), and 33 (77%) patients who achieved a binocular uncorrected distance, intermediate, and near visual acuity, respectively, better than 20/20 separately. Defocus curves revealed better distance vision in the dominant eyes, and the nondominant eyes performed better at intermediate and near vergence (P < .001). The Q-value and corneal spherical aberration coefficient were more positive in the dominant eyes than those in the nondominant eyes (P < .001). The accommodative amplitude and relative accommodation improved binocularly (P < .001). The questionnaire demonstrated high patient satisfaction with near vision, and no one reported having severe visual disturbance. For myopic patients with age-related accommodation deficiency, the Custom-Q algorithm proved to be an effective way to achieve acceptable near vision without compromising distance vision. [J Refract Surg. 2024;40(4):e245-e252.].

Electromagnetic lensing using the Aharonov–Bohm effect

We demonstrate theoretically and experimentally an electromagnetic lensing concept using the magnetic vector potential—in a region free of classical electromagnetic fields—via the Aharonov–Bohm (AB) effect. This toroid-shaped lens with poloidal current flow allows for electromagnetic lensing which can be tuned to be convex or concave with a spherical aberration coefficient of opposite polarity to its focal length. This field-free lens combines the advantages of traditional electromagnetic and electrostatic field-based lenses and opens up additional possibilities for the optical design of charged-particle systems. More generally, these results demonstrate that the AB effect can shape charged particle wavefronts beyond simple step shifts if topologies beyond simple flux lines are considered and further supports the physical significance of the magnetic vector potential.

Multi-focus manipulation system based on SNES aberration self-calibration

Abstract Wavefront shaping using digital micromirror devices (DMDs) allows inertia-free focus manipulation with numerous modulation modes and high refresh rates. However, the aberration caused by the curvature of DMD affects the focusing performance. Here, we propose an aberration self-calibration method based on separable natural evolution strategies (SNES). This method searches optimal Zernike coefficients of aberration globally and completes compensation using super-pixel encoding. Compared to the genetic algorithm method, we improve the speed by 62% and achieve better-focused spots. It enables simultaneous scanning of 25 independent focal spots. This advancement supports wavefront shaping applications in optical imaging, industrial inspection, and laser processing.

Optical systems with translational invariance.

The leading order monochromatic aberrations are investigated for optical systems, which obey single-plane symmetry and translational invariance. These aberrations are classified from the symmetry principles for the wave aberration function. Fermat's principle is applied for a system made of generic cylindrical surfaces, and a complete set of the aberration coefficients required for calculation of the aberrations for any paraxial ray is obtained. To demonstrate the applied value of the analytical results obtained, the criteria of compensation of the leading aberrations for the cylindrical analog of the Cassegrain telescope were found.

Design of a Spaceborne, Compact, Off-Axis, Multi-Mirror Optical System Based on Freeform Surfaces

Based on the application requirements of high spectral resolutions, high spatial resolutions and wide swatches, a new-generation, high-performance, spaceborne, hyperspectral imaging spectrometer (NGHSI) with a spatial resolution of 15 m and a swatch of 90 km is proposed. The optical system of the NGHSI has a focal length of 1128 mm, an F-number of three, a field of view (FOV) of 7.32° and a slit length of 144 mm. A new off-axis, multi-mirror telescope structure with intermediate images is put forward, which solves the design problem that realizes secondary imaging and good telecentricity at the same time. And a new off-axis lens-compensation Offner configuration is adopted to address the challenge of the high-fidelity design of spectral imaging systems with long slit lengths. The relationship between X-Y polynomials and aberration coefficients is analyzed, and the X-Y polynomial freeform surfaces are used to correct the off-axis aberrations. The design results show that the image quality of the telescope system is close to the diffraction limit. The smile, known as the spectral distortion along the line, and keystone, which is the magnification difference for different wavelengths, of the spectral imaging system are less than 1/10 pixel size. The complete optical system of the NGHSI, including the telescope system and the spectral imaging system, has excellent imaging quality and the layout is compact and reasonable, which realizes the miniaturization design.

A high-precision and efficient adaptive alignment method for Cassegrain dual mirror optical system based on machine learning algorithms

The Cassegrain telescope is widely used in aerospace exploration equipment, characterized by compact structure, complex optical path, and high imaging quality. However, due to the difficulty in establishing an accurate correspondence between the relative pose and imaging quality of a multi mirror group with real machining errors, the current Cassegrain telescope assembly process is very difficult, with blind operation, time-consuming, and low accuracy. This article proposes a high-precision adaptive alignment method for Cassegrain dual mirror optical systems based on machine learning algorisms, and an experimental adaptive alignment system with uncoupled degrees of freedom is developed. Firstly, the overall architecture of the adaptive alignment method is proposed consists of detection, calculation and alignment modules. In the detection module, the Zernike polynomial coefficient of wavefront aberration of the optical system is detected online through the interferometer, meanwhile the pose coordinates of the secondary mirror is accurately fed back through a 6-DOF nanoscale micro mechanism. In the calculation module, machine learning algorithm is applied to build a nonlinear mapping model between the Zernike coefficient and the pose coordinates of the secondary mirror. In the alignment module, the pose coordinates of the secondary mirror can be forced to adjust to the target position. Then, during the real alignment process, the Zernike coefficient test data of the optical system alignment process is monitored in real time, and the nonlinear mapping model is employed to calculate the pose coordinates and then the misalignment of the secondary mirror. Finally, the alignment module is driven to execute the pose correction according to the calculated misalignment value, realizing a high-precision adjustment of the Cassegrain dual mirror optical system. Experimental results shows that the average alignment time cost can be dramatically reduced from around 7 days using the current manual alignment method to just 30 minutes using the proposed adaptive alignment method for achieving a current standard alignment accuracy of wavefront aberration root mean square (RMS) less than 0.1λ, which greatly improves the assembly efficiency. This study proposes a new method for high-precision and efficient alignment of optical systems based on artificial intelligence and contributes to the efficiency improvement for optical assembly.

Design and optimization of a conical electrostatic objective lens of a low-voltage scanning electron microscope for surface imaging and analysis in ultra-high-vacuum environment

Low-voltage scanning electron microscopy (LV-SEM) with landing energies below 5 keV has been widely used due to its advantages in mitigating the damage and charging effects to a specimen and enhancing surface information due to small interaction volume of electrons inside a specimen. Additionally, for elemental analysis of the surfaces of bulk specimens with Auger electron spectroscopy (AES) or electron energy loss spectroscopy (EELS), ultra-high-vacuum (UHV) environment is essential to maintain clean surfaces without the absorption of gas molecules during the electron beam irradiation for the acquisition of spectral data. In this study, we propose the optimal design and condition of a conical Electrostatic Objective Lens (EOL) for a UHV LV-SEM to achieve the high spatial resolution and secondary electron (SE) detection efficiency. The EOL is composed of only the three electrodes (retarding, focusing and booster electrodes) and the insulators, which is suitable for maintaining a UHV environment with less out-gassing. The cone angle of the EOL is determined as 60° to integrate a spectrometer in the UHV LV-SEM and in a large size and a higher tilt angle of the sample. Through the optimization with the simulations, the EOL achieves the minimized spherical and chromatic aberration coefficients of 0.05 and 0.03 mm at the sample side, respectively, at the landing energy of 50 eV and the shortest working distance (WD) of 1 mm for high-resolution imaging. In addition, the probe diameter of the optimized EOL is 2.3 nm at 1 keV and 5.7 nm at 50 eV with a WD of 1 mm and a probe current of 10 pA, which are comparable to previously studied compound objective lenses with magnetic and electrostatic lenses. Using a longer WD of 4 mm for analysis, the probe diameter was 5.4 nm at 1 keV and the SE detection efficiency was 83.3 % owing to the separated scintillator detector structure from the booster electrode.These results imply that the optimized EOL has the potential to be applied to a high-performance UHV LV-SEM for the surface imaging and analysis with a simple system configuration.

Optimizing the Effectiveness of Magnetic Lenses by utilizing the Electron Optical Design (EOD) Software

This paper introduces a computational analysis that discusses an approach for optimal synthesis in the design of magnetic lenses, specifically focusing on the analytical method. A widely employed approach for magnetic lens design involves utilizing an analysis optimization procedure, which makes use of the finite element method and is supported by Munro programs. In this study, this approach has been employed to explore magnetic lenses using the Electron Optical Design (EOD) software. The study offers insights into the role of the air gap in magnetic lens design, highlighting its importance in optimizing objective and projector properties. The analysis reveals that variations in the air gap (S) significantly influence the performance of magnetic lenses. Decreasing the air gap when it is set to (3) leads to substantial improvements in objective optical properties and focal length. Conversely, increasing the air gap when it is set to (12) enlarges the half-width of the axial magnetic field while reducing the maximum magnetic field value. These findings underscore the importance of carefully optimizing the air gap to achieve desired lens performance. The focal length is determined using this input data and coefficients of aberration (spherical and chromatic) of the objective. The study focuses on the influence of a crucial geometric parameter, specifically the air gap (S), on both objective and projector properties. Its importance stems from its capability to pinpoint the suitable geometry for magnetic lenses, thereby facilitating their efficient application.

Optical Behavior of an Enhanced Monofocal Intraocular Lens Compared with a Standard One

The aim of this work was to compare an enhanced monofocal (RayOne EMV RAO200E, Rayner) and standard monofocal (RayOne RAO600C Aspheric, Rayner) intraocular lenses (IOLs) for three nominal powers (+10.00 D, +20.00 D and +30.00 D) as a function of the optical aperture diameter (pupil diameter) using a commercial Schlieren phase-shifting deflectometer NIMO TR1504 (Lambda-X, Belgium). From the wavefront maps measured by this instrument, the radial power profiles, the spherical aberration coefficients of the Zernike polynomial expansion (as a function of the optical aperture radius), and the root-mean-square (RMS) of the high-order aberrations (HOAs) were obtained and analyzed by comparing the two models. The results showed that the effective added power that could be obtained with the enhanced model depended directly on the pupil size and the power of the IOL implanted. The higher additions were achieved with the higher nominal IOL powers. The relationship between the pupil diameter, the corneal aberration of the patients and the power profile of these IOLs could have a crucial implication on the far distance and the final effective addition. However, it is important to note that these findings should be clinically validated through the implantation of these models in patients’ lenses.

Polarization aberration analysis and suppression of an infrared remote sensing optical system based on a digital micro-mirror device

The off-axis polarized infrared remote sensing imaging system based on a digital micro-mirror device (DMD) is capable of obtaining super-resolution images and enhancing the acquisition of target details. Polarization aberrations can reduce system resolution and affect target detection accuracy, it is more obvious in infrared imaging optical systems with small F-numbers and large incidence angles. It is complex to analyze and suppress the inherent polarization aberrations of infrared optical systems. To address this challenge, we propose a simplified polarization analysis method to get the aberration coefficients by using the specific pupil’s aberration value. The ray sampling rate of simplified method is 0.02 times that of traditional method, and the accuracy is almost unchanged. A method using weak polarization elements to suppress polarization aberrations is developed. Compared to the values before suppression, there is a reduction of more than 50% in diattenuation and retardance under different wavelength and fields. The shift of PSF centroid reduces over 55%, and the ellipticity of PSF decreases over 40%. Furthermore, the implementation of this technique has resulted in 28% improvement in the accuracy of measuring the degree of linear polarization (DoLP) of the optical system.

First- and third-order chromatic aberrations in Glaser magnetic lens for object magnetic immersion

In this paper, the Gaussian optical properties and the first- and third-order isotropic and anisotropic chromatic aberrations of the Glaser magnetic lens have been derived analytically and then calculated numerically based on two methods: the differential-algebraic DA and the electron optical aberration integrals. These coefficients have been recalculated for object magnetic immersion OMI. The expressions of chromatic aberration coefficients with the inclusion of the OMI effect of round electron lenses were not published in the literature until the present work. The numerical results of high-order chromatic aberration coefficients of Glaser magnetic lenses calculated using the DA method are shown to be in excellent agreement with those calculated using the integral aberration method and with the minor relative errors of order (10−7- 10−8), indicating that all the formulas for the chromatic aberration coefficients are entirely correct. For this purpose, COSY INFINITY 10 and Mathematica 11 were used, and both proved to be excellent computer programs for this work.