Vector Calculation Research Articles

Ab initio investigation of structural, elastic, dynamic, and electronic properties of YN binary rare earth nitride in ZB cubic phase

This research employs ab initio calculations, utilizing the FP-LAPW method as implemented in WIEN2k and the PP-PW method in QUANTUM ESPRESSO, within the framework of the GGA-PBE approximation, to investigate the physical properties of the rare earth nitride binary YN. Structural parameters of YN are determined, yielding predictive results consistent with recent calculations. Furthermore, a comprehensive investigation into mechanical behavior and elastic properties confirms the mechanical stability of YN in its cubic ZB structure. An analysis of the anisotropy coefficient reveals its elastic anisotropy. The dynamic stability of YN is investigated through PW-scf calculations using DFPT theory, enabling the calculation of phonon spectra, frequencies, and polarization vectors. These findings affirm the stability of YN material in the B3 structure, which can be experimentally synthesized. Electronic properties are probed, showing semiconductor behavior, utilizing GGA mBJ, and YS PBE0 methods to ascertain band gap values in the B3 structure. This suggests its potential for adoption in optoelectronic and photonic devices.

Seesaw: A 4096-bit vector processor for accelerating Kyber based on RISC-V ISA extensions

The ML-KEM standard based on Kyber algorithm is one of the post-quantum cryptography (PQC) standards released by the National Institute of Standards and Technology (NIST) to withstand quantum attacks. To increase throughput and reduce the execution time that is limited by the high computational complexity of the Kyber algorithm, an RISC-V-based processor Seesaw is designed to accelerate the Kyber algorithm. The 32 specialized extension instructions are mainly designed to enhance the parallel computing ability of the processor and accelerate all the processes of the Kyber algorithm by thoroughly analyzing its characteristics. Subsequently, by carefully designing hardware such as poly vector registers and algorithm execution units on the RISC-V processor, the support of microarchitecture for extension instructions was achieved. Seesaw supports 4096-bit vector calculations through its poly vector registers and execution unit to meet high-throughput requirements and is implemented on the field-programmable gate array (FPGA). In addition, we modify the compiler simultaneously to adapt to the instruction extension and execution of Seesaw. Experimental results indicate that the processor achieves a speed-up of 432× and 18864× for hash and NTT, respectively, compared with that without extension instructions and a speed-up of 5.6× for the execution of the Kyber algorithm compared with the advanced hardware design.

Формування траєкторії керованих літальних апаратів

Currently, there is an increased interest in the creation of flight control systems for unmanned aerial vehicles, which, taking into account real conditions, would ensure its trajectory and thereby determine the effectiveness of the use of the aerial vehicle. Aircraft navigation, in particular ballistic cruise missiles, requires the formation of high-precision control in order to achieve both final results and local optimization tasks at each of the flight stages: take-off, departure to set trajectories, maneuvering, etc. The development of admissible control and its optimization by computer-mathematical methods of modeling and optimization is a time-consuming process and requires significant expenditures of various types of resources. An element of the effectiveness of such a mathematical apparatus for guaranteeing and increasing the reliability and effectiveness in achieving the set goal is the speed of calculations, which, in turn, requires the simplification of the mathematical model by obtaining functional dependencies for calculating the flight trajectory, avoiding complex mathematical calculations. The complete working model for calculating the desired trajectory of the aircraft will be determined by the specified control functions and will be provided by a system of partial differential equations with time, and the necessary parameters for calculating all aerodynamic forces and moments are tabular data in most cases. A mathematical model for solving the task of forecasting navigation for such complex controlled systems is proposed to be carried out using two coordinate systems: starting and speed. Moreover, all calculations of the velocity vector, which depends on the net effect of three aerodynamic forces acting on the aircraft, should be carried out in the speed coordinate system, and coordinate control of the flight along the entire dynamic trajectory should be determined in the starting system

Uncertainty analysis for free-space three-dimensional measurement of electromagnetic pulse

As the polarization direction is unknown in free-space three-dimensional (3D) electromagnetic pulse (EMP) measurement, the components in three directions are usually measured first, and then the total vector is calculated. The waveform (magnitude) and polarization direction define the 3D process. Because of the uncertainty produced in the component measurement, there is also uncertainty in the calculated 3D process. This paper investigates the propagation of uncertainty during the total vector calculation process. The magnitude and polarization angle uncertainty propagation formulas are derived through analysis. The results show that the uncertainty of the calculated magnitude is less than the maximal measurement uncertainty of the three components, and the uncertainty of the polarization angle is less than sqrt(2) times the maximal uncertainty of the three components divided by the magnitude of the measured field. Finally, a Monte-Carlo (MC) simulation is run to validate the results of the analysis. The simulation results agree well with the analysis results.

Fast and accurate distal locking of interlocked intramedullary nails using computer-vision and a 3D printed device

IntroductionDistal locking is a challenging and time-consuming step in interlocked intramedullary nailing of long bone fractures. Current methods have limitations in terms of simplicity, universality, accuracy, speed, and safety. We propose a novel device and software for distal locking using computer vision.Methods and materialsThe device consists of an universal ancillary clamp, a telescopic arm, a viewfinder clamp, and a radio-opaque cross. The software uses a camera photo from the C-arm intensifier and adjusts for geometric projection deformities. The software employs edge detection, Hough transform, perspective interpolation, and vector calculation algorithms to locate the distal hole center. The device and software were designed, manufactured, and tested using 3D CAD, FEM, DRR, and performance testing on phantom bones.ResultsThe device and software showed high accuracy and precision of 98.7% and 99.2% respectively in locating the distal hole center and calculating the correctional vector. The device and software also showed high success ratio in drilling the hole and inserting the screw. The device and software reduced the radiation exposure for the surgeon and the patient. The success ratio of the device and software was validated by the physical testing, which simulated the real clinical scenario of distal locking. The radiation exposure was as low as 5 s with a radiation dose of 0.2mSv, drastically reducing radiation exposure during distal locking.DiscussionOur device and software have several advantages over other distal locking methods, such as simplicity, universality, accuracy, speed, and safety. Our device and software also have some disadvantages, such as reliability and legislation. Our device and software can be compared with other distal locking methods based on these criteria. Our device and software have some limitations and challenges that need to be addressed in the future, such as clinical validation, and regulatory approval.ConclusionThe device showed promising results in terms of low-cost, reusability, low radiation exposure, high accuracy, fast distal locking, high stiffness, and adaptability. The device has several advantages over other distal locking techniques, such as free-hand technique, mechanical aiming devices, electromagnetic navigation systems, and computer-assisted systems. We believe that our device and software have the potential to revolutionize the distal locking technique and to improve the outcomes and quality of life of the patients with long bone fractures.

Prediction of Visual Acuity in Pseudophakic Cataract Population Based on Residual Refraction

Purpose The purpose of the study was to design a simple, handy prediction for the effect of spherical and cylindrical refractive error on the visual acuity degradation at different distances and validate this model on a clinical dataset. Methods This study examined 70 eyes from 35 patients’ post-cataract surgery with aberration-free intraocular lenses. Biometric and corneal data were analysed, and subjective refraction and visual acuity were evaluated by two experienced optometrists. The study computed the spherical equivalent (SEQ), and defocus equivalent via vector addition (DEQ vec), as the sum of absolute values (DEQ abs). Predictive models were developed using univariate regression, with confidence intervals (BCa 95%) calculated through non-parametric bootstrapping (10,000 cycles). Results Various calculated equivalents included −0.44 D for spherical equivalent (SEQ), 0.70 D for defocus equivalent based on vector calculation (DEQ vec), and 0.89 D for defocus equivalent based on absolute values (DEQ abs). Uncorrected and corrected visual acuity averaged 0.07 logMAR and −0.04 logMAR, respectively. The absolute defocus equivalent (DEQ abs) exhibited the smallest confidence interval (BCa 95%) at 0.07. Conclusion The defocus equivalent based on the addition of absolute values (DEQ abs) emerged as the most practical predictor for the described applications. Notably, it offers the advantage of easy calculability through a simple equation: VA loss = DEQ abs ⋅ 0.23. In 95% of cases, this predicted loss would have an accuracy of ±0.03 lines.

High-efficiency sound source localization using data-driven sparse sampling with validation using monopole laser sound source

This study proposes a framework that reduces the calculation cost of sound source localization with the Amiet model, using a data-driven sparse sampling method. This method accelerates the calculation of the steering vector used in conventional beamforming. An aeroacoustic wind tunnel test was conducted in a 2 × 2 m2 low-speed wind tunnel, and the proposed method was verified. During the test, a monopole laser sound source, which does not interfere with the flow, was used, and its acoustic signals were measured using a microphone array. Next, steering vectors were reconstructed by discovering dominant modes and optimized sampling points from the training data based on the Amiet model and the modified data-driven sparse sampling method. Finally, the sound-source positions when the steering vector of the proposed model was used were compared with the positions observed when the steering vector of which all the grid points were calculated was used. The error was less than 2 mm when 16 modes were used, and the calculation time was reduced to ∼1/33 of that of the previous Amiet model.

Scalable Analysis of Dipole Moment Fluctuations for Characterizing Intermolecular Interactions and Structural Stability.

Accurately predicting protein-ligand interactions is essential in computational molecular biochemistry and in silico drug development. Monitoring changes in molecular dipole moments through molecular dynamics simulations provides valuable insights into dipole-dipole interactions, which are critical for understanding protein structure stability and predicting protein-ligand binding affinity. In this study, we propose a novel method to monitor changes in the interangle between dipole vectors of residue molecules within proteins and ligand molecules, aiming to evaluate the strength and consistency of interactions within the complex. Additionally, we extend the concept of positional root-mean-square fluctuation (RMSF), commonly used for protein structure stability analysis, to dipole moments, thus defining dipole moment RMSF. This enables us to analyze the stability of dipole moments for each residue within the protein and compare them across residues and between binding and non-binding complexes. Using the CRBP1-retinoic acid complex as our model system, we observed a significant difference in the interangle change of dipole moments for the key residue at the residue-level between the non-binding and binding complexes. Furthermore, we found that the dipole moment RMSF value of the non-binding complex was substantially larger than that of the binding complex, indicating greater dipole moment instability in the non-binding complex. Leveraging the concept of scalability inherent in the calculation of dipole moment vectors, we systematically expanded the residues within the protein's primary secondary structure. Our dipole moment analysis approach can provide valuable predictive insights into complex candidates, especially in situations where experimental comparisons are challenging.

Efficient Gene Expression Data Analysis using ES-DBN For Microarray Cancer Data Classification

INTRODUCTION: DNA microarray has become a promising means for classification of various cancer types via the creation of various Gene Expression (GE) profiles, with the advancement of technologies. But, it is challenging to classify the GE profile since not all genes contribute to the presence of cancer and might lead to incorrect diagnoses. Thus an efficient GE data analysis for microarray cancer data classification using Exponential Sigmoid-Deep Belief Network (ES-DBN) is proposed in this work.OBJECTIVES: The study aims to develop an efficient GE data analysis using Exponential Sigmoid-Deep Belief Network (ES-DBN) for microarray cancer data classification.METHODS: The proposed methodology starts with pre-processing to compact data. Afterward, by utilizing Min-Max feature scaling technique, the pre-processed data is normalized. The normalized data is further encoded and feature ranking is performed. The subset values are selected using Cauchy Mutation-Coral Reefs Optimization (CM-CRO) in feature ranking. The feature vector is calculated by Pearson Correlation Coefficient based GloVe (PCC-GloVe) algorithm since different subsets return the same fitness value. Statistical and Biological validations take place after feature vector calculation. Lastly, for effective classification of the type of cancer, the vector features obtained are fed to ES-DBN.RESULTS: The outcomes of the proposed technique are evaluated with various datasets, which exhibited that the proposed technique performed well with the Ovarian cancer dataset and outperforms other conventional approaches.CONCLUSION: This study presents a comprehensive methodology for efficiently classifying cancer types using GE profile. The proposed GE data analysis using ES-DBN shows promising results, highlighting its potential as a valuable tool for cancer diagnosis and classification.

Improved stiff string torque and drag prediction using a computationally efficient contact algorithm

ABSTRACT Due to the intermittent contact with the wellbore, determining torque and drag for deviated wells is difficult. Most models have ignored drill string stiffness and assumed continual contact to simplify derivation. However, the accuracy of these ‘soft-string’ models is restricted, especially at high dogleg severities. On the other hand, most ‘stiff-string’ models rely on computationally intensive approaches or continuous contact assumptions. To mitigate these issues, a computationally efficient penalty-based wellbore contact algorithm has been developed based on vector calculation, which at most requires two dot products and three arithmetic operations to determine contact locations. This algorithm is incorporated into a 3D multibody dynamics (MBD) model, which utilizes rigid drill-string segments based on the Newton-Euler formulation, connected via axial, shear, torsional, and bending springs to capture drill string flexibility. This model performs simulations faster than real-time and has been validated using surface measurements from a completed well.

Refractive Astigmatism Outcomes of Femtosecond Laser-Assisted Arcuate Keratotomies Combined with Femtosecond Laser-Assisted Cataract Surgery: Two-Year Results

Purpose To evaluate the effectiveness and stability of refractive astigmatism reduction after penetrating femtosecond laser-assisted arcuate keratotomy performed at the time of femtosecond laser-assisted cataract surgery. Methods Non-randomized retrospective data analysis of all patients that underwent femtosecond laser-assisted cataract surgery with femtosecond laser-assisted arcuate keratotomy over a 4-year period with a non-toric monofocal intraocular lens (2017–2021) at a tertiary care academic center. Postoperative visual acuity, manifest refraction, and predicted residual refractive error were also recorded at 1 month, 3–6 months, 12–18 months, and 2 years postoperatively. Preoperative keratometric astigmatism was compared to postoperative refractive astigmatism using vector calculations and the ASCRS double-angle plot tool. Results This study comprised 266 eyes (179 patients) that met inclusion criteria. The mean preoperative keratometric astigmatism magnitude was 0.99 ± 0.53 D. At 1 month, 3–6 months, 12–18 months, and 2 years postoperatively, the mean refractive cylinder was 0.49 ± 0.45 D, 0.49 ± 0.45 D, 0.55 ± 0.54 D, and 0.52 ± 0.46 D, respectively. Horizontal against-the-rule astigmatism showed a higher tendency toward undercorrection than vertical with-the-rule astigmatism, which had a slightly higher tendency toward overcorrection. With-the-rule astigmatism had smaller difference vectors between target-induced astigmatism and surgically induced astigmatism. Conclusions Femtosecond laser-assisted arcuate keratotomy performed at the time of femtosecond laser-assisted cataract surgery was an effective option for correcting low-to-moderate corneal astigmatism for up to 2 years.

Optimal Frequency determination for Wind-Hydro hybrid unit using Telegrapher’s Method for Low frequency transmission lines

This study addresses the optimal load flow problem for low-frequency transmission lines connected to wind farms and pumped storage Hydropower (PSH). The losses in LF-HVac system and the voltage stability index heavily dependent on the operating frequency. Therefore, an optimal load flow (OLF) method must be incorporated to determine the proper solutions for these systems. This system analyses the load flow solutions in which newton Raphson with STATCOM requires large vector calculations with variables. So, to reduce the mathematical calculation a two-port equivalent model for transmission lines based on solutions to Telegrapher’s method has been proposed. The proposed OLF method is tested with other algorithms and implemented in IEEE 39-BUS system. The results obtained indicate that the LF method with the proposed metaheuristic algorithm gives optimal frequency for low-frequency HVac systems while maintaining the stability of system.

What should we understand by the four-momentum of physical system?

It is shown that, in general, in curved spacetime, none of the known definitions of four-momentum correspond to the definition, in which all the system’s particles and fields, including fields outside matter, make an explicit contribution to the four-momentum. This drawback can be eliminated under the assumption that the primary representation of four-momentum is the sum of two nonlocal four-vectors of the integral type with covariant indices. The first of these four-vectors is the generalized four-momentum, found with the help of Lagrangian density. The time component of the generalized four-momentum, in theory of vector fields, is proportional to the particles’ energy in scalar field potentials, and the space component is related to vector field potentials. The second four-vector is the four-momentum of fields themselves, and its time component is related to the energy given by tensor invariants. As a result, the system’s four-momentum is defined as a four-vector with a covariant index. The standard approach makes it possible to find the four-momentum in covariant form only for a free point particle. In contrast, the obtained formulas for calculating the four-momentum components are applied to a stationary and moving relativistic uniform system, consisting of many particles. In this case, the main fields of the system under consideration are taken into account, including the electromagnetic and gravitational fields, the acceleration field and the pressure field. All these fields are considered vector fields, which makes it possible to unambiguously determine the equations of motion of the fields themselves and the equations of motion of matter in these fields. The formalism used includes the principle of least action, charged and neutral four-currents, corresponding four-potentials and field tensors, which ensures unification and the possibility of combining fields into a single interaction. Within the framework of the special theory of relativity, it is shown that due to motion, the four-momentum of the system increases in proportion to the Lorentz factor of the system’s center of momentum, while in the matter of the system the sum of the energies of all fields is equal to zero. The calculation of the integral vector’s components in the relativistic uniform system shows that the so-called integral vector is not equal to four-momentum and is not a four-vector at all, although it is conserved in a closed system. Thus, in the theory of relativistic vector fields, the four-momentum cannot be found with the help of an integral vector and components of the system’s stress-energy tensor, in contrast to how it is assumed in the general theory of relativity.

3D Slicer open-source software plug-in for vector-based angle calculation of canine hind limb alignment in computed tomographic images.

Severe and complex angular limb deformities in dogs require accurate morphological assessment using diagnostic imaging to achieve successful orthopedic surgery. Computed tomography (CT) is commonly used to overcome projection errors in two-dimensional angular measurements of dog hindlimb alignment. Three-dimensional volume rendering (VR) techniques permit virtual positioning and variable projection, but the final CT-image that defines the projection plane for angular measurements remains two-dimensional. We wanted to develop a true three-dimensional open-source technique to measure the alignments of the hind limbs of dogs in CT scanners. We developed an open-source 3D Slicer plug-in, to perform angular measurements using vector calculations in three-dimensional space. In 113 CT-scans of canine pelvic limbs, femoral torsion, femoral varus, femorotibial rotation, tibial torsion, tibial varus and tibiotalar rotation angles were calculated and compared to an already validated technique using VoXim®. Reference points were identified and measurements were possible in the 113 acquisitions. The greatest difference between the two techniques was 1.4° at only one tibial torsion angle. Mean values for all Bland-Altman plots did not show significant differences and were less than 0.07° for all comparisons. Based on these results we considered angular measurements of canine hind limb alignment in CT scans using the 3D Slicer extension program sufficiently accurate for clinical orthopedic and surgical purposes in veterinary medicine. With our open-source 3D Slicer extension software, we provide a free accessible tool for veterinary orthopedic surgeons and thus we hope to improve angular measurements in CT-scans of canine hind limb deformities through true three-dimensionality.

A fast gradient convolution kernel compensation method for surface electromyogram decomposition

Decomposition of EMG signals provides the decoding of motor unit (MU) discharge timings. In this study, we propose a fast gradient convolution kernel compensation (fgCKC) decomposition algorithm for high-density surface EMG decomposition and apply it to an offline and real-time estimation of MU spike trains. We modified the calculation of the cross-correlation vectors to improve the calculation efficiency of the gradient convolution kernel compensation (gCKC) algorithm. Specifically, the new fgCKC algorithm considers the past gradient in addition to the current gradient. Furthermore, the EMG signals are divided by sliding windows to simulate real-time decomposition, and the proposed algorithm was validated on simulated and experimental signals. In the offline decomposition, fgCKC has the same robustness as gCKC, with sensitivity differences of 2.6 ± 1.3 % averaged across all trials and subjects. Nevertheless, depending on the number of MUs and the signal-to-noise ratio of signals, fgCKC is approximately 3 times faster than gCKC. In the real-time part, the processing only needed 240 ms average per window of EMG signals on a regular personal computer (IIntel(R) Core(TM) i5-12490F 3 GHz, 16 GB memory). These results indicate that fgCKC achieves real-time decomposition by significantly reducing processing time, providing more possibilities for non-invasive neuronal behavior research.

A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition

A curved non-isoparametric Reissner–Mindlin shell element is developed for analyzing thin-walled structures. The standard kinematic description of the element requires the calculation of the director vector. To address this demand accurately, similar to isogeometric analysis (IGA), the geometry is defined by utilization of the non-uniform rational B-splines (NURBS) imported directly from computer-aided design (CAD) files. Then, shape functions of the Legendre spectral element method (SEM) are used to interpolate the displacements. Consequently, the shell director vector and Jacobian of the transformation are calculated properly according to the presented formulation. On the other hand, in Legendre SEM combined with Gauss–Lobatto–Legendre quadrature, the integration points and the element nodes coincide. Thus, the easily computable local coordinate systems at the integration points can be used directly as nodal basis systems. A separate calculation of nodal basis systems at control points, which is the source of either complexity or error in IGA shells, is not required. Given the condition number of the stiffness matrix in the developed method, super high-order elements can also be used. Very high order p-refined elements are used in addition to h-refinement of the mesh to show the capability of higher order elements to analyze problems without mesh refinement. The validity and convergence rate of the method are investigated and verified through various cases of h- and p-refinement in challenging obstacle course problems.

New marker for chronic kidney disease progression and mortality in medical-word virtual space

A new marker reflecting the pathophysiology of chronic kidney disease (CKD) has been desired for its therapy. In this study, we developed a virtual space where data in medical words and those of actual CKD patients were unified by natural language processing and category theory. A virtual space of medical words was constructed from the CKD-related literature (n = 165,271) using Word2Vec, in which 106,612 words composed a network. The network satisfied vector calculations, and retained the meanings of medical words. The data of CKD patients of a cohort study for 3 years (n = 26,433) were transformed into the network as medical-word vectors. We let the relationship between vectors of patient data and the outcome (dialysis or death) be a marker (inner product). Then, the inner product accurately predicted the outcomes: C-statistics of 0.911 (95% CI 0.897, 0.924). Cox proportional hazards models showed that the risk of the outcomes in the high-inner-product group was 21.92 (95% CI 14.77, 32.51) times higher than that in the low-inner-product group. This study showed that CKD patients can be treated as a network of medical words that reflect the pathophysiological condition of CKD and the risks of CKD progression and mortality.

Flow Visualisation by Laser Sheet in a Smoke-Tunnel

The application of wind tunnel for visualizing flow over any bluff or streamline body plays an indispensable role in concepting various aerodynamical phenomenon such as formation of vortices, generating lift and drag, calculation of velocity vector and pressure profiles etc. Flow visualization through smoke and laser optics in a subsonic wind tunnel to capture vortex formation over airfoils, cylindrical and conical bodies is the experimental approach which has been studied and manifested elaborately in the article. Using smoke as a seeding material, the flow of the same is allowed to pass over the model which is illuminated via a double pulse laser beam to visualize the formation of vortex and capture the same for future references. The experimentation of the above models is carried out in a Honeycomb subsonic wind tunnel in the Aerodynamics Lab of Aerospace Engineering in the university. The article will cover Computational Fluid Dynamics (CFD) performed on ANSYS for the particular NACA series, Cross correlation PIV for plotting velocity vector for the captured images of the seed particles and the respective images of the entire experimental set-up. The images captured during this experiment can be later used by the aerodynamicists for optimizing aerodynamic efficiency of the experimented models depending on the vortex formation and flow pattern. The images can also be used for advance calculations required in Particle Image Velocimetry (PIV) for calculating velocity of every speed particle.

Numerical simulation of dry laser derusting process based on SPH method

Laser cleaning is an efficient, environmental-friendly, and non-contact surface treatment technology. Laser radiation and heating cause the surface material to heat up and gasify, separating it from the substrate. Traditional mesh-based numerical methods are difficult to effectively simulate the evolution of erosion crater and the splashing phenomenon. In this study, a meshfree method, the smoothed particle hydrodynamics (SPH) method, is used to establish the numerical model of the interaction between the laser beam and the targeted material and to investigate the removal process of the rust layer under different laser operating conditions. In consideration of the coupling effect of laser absorption, heat transfer, and material phase change, the SPH modeling procedure and corresponding numerical scheme for heat transfer and heat-absorption-induced phase change are introduced. Additionally, a surface particle detection algorithm and surface normal vector calculation method are proposed to accurately compute the complex surface geometry of the erosion crater, which realizes the dynamic coupling of laser-energy absorption and laser-beam direction. The established SPH model is then used to simulate the temperature distribution of the rust layer under the action of a laser beam, and the influence of laser energy, beam overlap rate, and beam direction on the removal efficiency is analyzed. This study applies the meshfree SPH method to the study of laser rust removal process, verifies the accuracy of the surface detection algorithm, captures the spatter behavior of material particles after phase change, and reflects the advantages of the meshfree method in solving such problems.

Sigma vector calculations in nodal aberration theory and experimental validation using a Cassegrain telescope

Nodal Aberration Theory (NAT) was developed to explain the field dependency of aberration field centers in the image plane of nominally rotationally symmetric optical systems that have lost their symmetry through misalignments. A new insight into the theory led to calculating the sigma vectors, which locate the aberration field centers, using the angle between a real-ray trace of the optical axis ray (OAR) and the normal of the local surface where “local” refers to the object and image optical spaces of that surface. Here, we detail the sigma vector calculations for general optical systems and provide an experimental investigation of a misaligned system with a high-precision customized Cassegrain telescope. In the simulations, a Newtonian telescope, a Cassegrain telescope, and a three-mirror anastigmat telescope were misaligned intentionally in ray-tracing software. The sigma vectors were calculated analytically for the third-order aberrations of astigmatism and coma. Experimentally, the same perturbations were implemented for the Cassegrain telescope system, and the aberrations were quantified through interferometric measurements on a grid of field points in the image plane that verified the analytical derivation and simulations.