Beamwidth Research Articles

Wakefield-driven filamentation of warm beams in plasma.

Charged and quasineutral beams propagating through an unmagnetized plasma are subject to numerous collisionless instabilities on the small scale of the plasma skin depth. The electrostatic two-stream instability, driven by longitudinal and transverse wakefields, dominates for dilute beams. This leads to modulation of the beam along the propagation direction and, for wide beams, transverse filamentation. A three-dimensional spatiotemporal two-stream theory for warm beams with a finite extent is developed. Unlike the cold beam limit, diffusion due to a finite emittance gives rise to a dominant wave number and a cutoff wave number above which filamentation is suppressed. Particle-in-cell simulations with quasineutral electron-positron beams in the relativistic regime give excellent agreement with the theoretical model. This paper provides deeper insights into the effect of diffusion on filamentation of finite beams, crucial for comprehending plasma-based accelerators in laboratory and cosmic settings.

Influence of Pumping Laser Intensity Uniformity on Hybrid Optically Pumped Comagnetometer in the SERF Regime

AbstractThe effect of pumping laser intensity uniformity (PLIU) on the operation of a hybrid optically pumped comagnetometer operating in the spin‐exchange relaxation‐free regime (SERF) regime is investigated in this paper. First, an analytical steady‐state output model for the comagnetometer with two alkali‐metal atoms and one noble‐gas atom is presented. By varying the diameter of the pumping laser beam to control the PLIU, 3D distribution models of the electron spin and nuclear spin polarization under different PLIU conditions are obtained. In the experiment, the effects of PLIU on multi‐parameter, low‐frequency magnetic‐noise suppression capability, and long‐term stability of the SERF comagnetometer are studied. The results indicate that within a certain range, increasing the diameter of the pumping laser beam improves the polarization uniformity of the atomic ensemble and reduces the light shift of the comagnetometer. As a result, both the low‐frequency magnetic noise suppression capability and the long‐term stability of the system increase. However, further reduction of the pumping laser diameter leads to a reversal of the system performance metrics, suggesting the presence of a tipping point. The research presented in this article is critical for advancing the efficient polarization study and hyperpolarization of SERF comagnetometers.

A frequency steerable electromagnetic acoustic transducer

Abstract Electromagnetic acoustic transducers (EMATs) are convenient for non-destructive evaluation of plate-like structures since they can generate, without the need for contact with the medium under test, different types of ultrasonic guided waves. Guided-wave EMATs usually generate waves omnidirectionally or in a principal propagation direction. Beam steering is desirable in several applications, such as in inspections of large-area structures. This is usually achieved with several independently controlled elements forming a phased array. Alternatively, mono-element transducers with directional-dependent spectral content can steer the generated wave beam by altering the frequency of the excitation signal. A piezoelectric transducer with this characteristic, namely a frequency steerable acoustic transducer, was previously proposed. Its design was addressed in the wavenumber domain, leading to unconventional transducer shapes, but still reproducible with a piezoelectric patch, albeit unfeasible to implement as an EMAT. Here, we propose a new kind of EMAT, namely, frequency steerable EMAT (FSEMAT), whose design is addressed in the spatial domain in order to ensure its physical realization with a coil-magnet arrangement whilst still effectively presenting steering capability. The novel EMAT was designed to generate the A 0 Lamb wave mode in a frequency range from approximately 100 to 600 kHz. The FSEMAT was fabricated and experimentally evaluated in an aluminium plate at different frequencies within the designed frequency range, where each frequency corresponded to a specific propagating direction with high directivity, assessed by half-power beam widths of approximately 10 degrees. Furthermore, its theoretical directivity was computed by means of a wavenumber spectrum-based model, and showed good agreement with experimental results. The new transducer allows great flexibility effectively providing beam steering with a single EMAT.

Optimization of polarization spectroscopy signal for laser-frequency stabilization

Abstract We experimentally generate polarization spectroscopy (error) signals corresponding to the D2-line hyperfine transitions, F g = 2 → F e = 1 , 2 , 3 of 87Rb, and F g = 3 → F e = 2 , 3 , 4 of 85Rb, and show that the strongest error signals correspond to the closed hyperfine transitions (oscillator strength ∼ 0.7 ), F g = 2 → F e = 3 and F g = 3 → F e = 4 , respectively. We make the generated error signal robust to fluctuations in external parameters by finding optimum values for the vapor cell temperature and pump-probe intensities at two different beam diameters. We further employ these optimized error signals to directly (without the need for frequency/phase modulation) lock the laser frequency—reducing the rms drift/linewidth from ∼ 10 MHz to < 500 kHz, when measured over a ∼ 60 min duration. In addition, by comparing theoretically calculated and experimentally measured error signals for the pump intensities ranging from ∼ 0.1 I sat 0 − 10 I sat 0 (where, I sat 0 ∼ 1.6 mW cm−2 is the two-level saturation intensity for the Rb D2-line transitions), we discuss the applicability and limitations of existing numerical and analytical approaches based on a full multi-level rate equation model for polarization spectroscopy.

Waveform of the Reflected Impulse at the Oblique Sounding of the Sea Surface

The height of sea waves is one of the most important characteristics describing the wave climate of the ocean. At the present, the main radar for remote measurement of wave heights is an altimeter. Measurements are performed at the vertical sounding (incidence angle equal to zero). The Brown model was developed to describe the waveform of the reflected impulse at the vertical sounding. There is no theoretical model for the case of oblique sounding. In the Kirchhoff approximation, the theoretical task about waveform of the reflected impulse at oblique sounding was considered. In the result of the investigation, the analytical formula for the waveform of the reflected impulse for oblique sounding at the small incidence angles (< 12◦) for a microwave radar with a narrow antenna beam was obtained. The waveform of the reflected impulse depends on the width of antenna beam, incidence angle, impulse duration, significant wave height (SWH), altitude of the radar, mean square slopes of large-scale, in comparison with radar wavelength, sea waves. It is shown that possibility exist to retrieve SWH using waveform the reflected impulse at the oblique sounding.

Cumulant method for identifying spatial distributions of laser beam intensity

The article considers the identification of spatial distributions of laser beam intensity, which is of significant practical importance for laser developers when certifying radiation sources. As an identification parameter, a measure of similarity of the measured axisymmetric distributions of laser radiation intensity with a given Gaussian distribution is proposed and investigated. To determine this measure, it is not necessary to first find the beam width at the waist. A mathematical apparatus for identification based on the use of cumulants of spatial intensity distributions is developed. A feature of the method is a limited number of cumulants of the Gaussian distribution, which makes it attractive for practical application. A comparative analysis of a number of two-dimensional distributions with a Gaussian distribution is carried out and it is shown that even with a small number of measured cumulants, high sensitivity of the method to the shapes of intensity distributions is ensured. A similarity measure for continuous one- dimensional intensity distributions of arbitrary shape is presented. The asymmetry measure of two-dimensional spatial distributions of laser beam intensity based on the ratio of moments and cumulants of the fourth and second orders is considered. The proposed measure of similarity of the measured axisymmetric intensity distributions with the Gaussian distribution is an alternative to the well-known propagation coefficient M 2 and can be used together with this coefficient.

A wide angle beam scanning CRLH leaky-wave antenna with non-identical elements per unit-cells

In this research article, a wide-angle scanning leaky-wave antenna (LWA) based on the composite right/left-handed (CRLH) metamaterial (MTM) operating over the C-band spectrum for mobile applications is proposed, investigated, and realized. The proposed LWA includes nine CRLH-MTM-based unit-cells. The properties of CRLH-MTM have been applied using series slots and shunt metallic via-holes. Each unit cell consists of three patches, two external patches, and one smaller middle patch implemented between them. One slot has been etched on each external patch; the width of these two slots is unequal. Moreover, the distance of the middle patch with each side patch is different. The results show that using this method of realizing non-identical elements caused the achievement of proper return-loss at the transition frequency and provided the balanced CRLH metamaterial condition. Additionally, using CRLH-MTM properties by realizing series capacitance (slot) and shunt inductance (via-holes) made the wide-angle scant from backward to forward radiation pattern. The results demonstrate that the LW antenna with small dimensions of 251.5 × 30 × 1.575 mm3 operates over a frequency range of 4.8–6.13 GHz covering a wide beam scanning range from −63° to +65° with broadside radiation at the center frequency of 5.46 GHz. The radiation gain of the proposed antenna varies from 6.2–13.1 dB. The achievements confirm the effectiveness of the proposed methods to realize a high-performance LW-antenna with the advantages of simple design, low profile, low cost, and easy of manufacture for mobile applications.

Optical Methods for Optimizing Fluorescence Imaging Field of View and Image Quality in Surgical Guidance Procedures.

Cancer surgery is aimed at complete tumor resection and accurate lymph node detection. However, numerous blood vessels are distributed within the tumor, and the colors of the tumor, blood vessels, and lymph nodes are similar, making observations with the naked eye difficult. Therefore, tumors, blood vessels, and lymph nodes can be monitored via color classification using an operating microscope to induce fluorescence emission. However, as the beam width of the LED required to induce fluorescence emission is narrow and the power loss of the beam is significant at a certain working distance, there are limitations to inducing fluorescence emission, and light reflection occurs in the observation image, obstructing the view of the observation area. Therefore, the removal of reflected light is essential to avoid missing the diagnosis of the lesion under observation. This paper proposes the use of a beam mirror and polarizing filter to increase the beam width and beam intensity. The refraction and reflection effects of the beam were utilized using the beam mirror, and the rotation angle of the polarizing filter was adjusted to remove light reflection. Consequently, the minimum beam power using the beam mirror was 10.9 mW, the beam width was doubled to 40.2°, and more than 98% of light reflection was removed at 90° and 270°. With light reflection effectively eliminated, clear observation of lesions is possible. This method is expected to be used effectively in surgical, procedural, and diagnostic departments.

Design of a metalens for beam collimation and angular amplification in optical phased array devices

We present an analytical design for increasing the beam sharpness (collimation) and field of view (FOV) of an optical phased array (OPA) device. In this work, a cylindrical metalens is used for collimation, while a set of metalens, with both concave and convex phase profiles, are incorporated to increase the FOV. Following the generalized vector law of reflection or refraction, the trajectories of the reflected or transmitted rays corresponding to the phase profile of phase gradient metasurfaces/metalens are obtained. Through the ray tracing method, the elliptical beam from the OPA device with a vertical beam (fast axis) width of 21 mm was collimated to a sharp spherical beam of width 1.5 mm by a metalens with a cylindrical phase profile. In addition, the incorporation of angular amplifier metalens with a 64-channel OPA device has shown an increased in FOV by almost threefold i.e., from 15 o to 41.96 o . Our results suggest that the use of metasurfaces/metalens can enhance the quality of output beam and provide significant advantages for compact on-chip integration with OPA devices in solid-state LiDAR applications.

Innovative K-band slot antenna array for radar applications

This article introduces a novel microstrip slot antenna array (SAA) configuration for radar applications. The proposed antenna is specifically designed for operation in the K-band, spanning from 23 to 24.3 GHz. The antenna structure comprises two substrates: the feed network and ground plane are on the bottom substrate, and the radiating slots are on the top layer of the first substrate. The incorporation of a unique grid feed configuration, featuring 50 Ohm center excitation for the first time, improves the feed mechanism of the microstrip SAA. This innovation contributes to achieving a compact size and high gain. To enhance the side lobe level, the design incorporates a substrate-integrated waveguide-backed cavity, which significantly reduces surface waves. The SAA consists of 25 radiating elements with a gain of 14 dBi. In the elevation and azimuth planes, the half-power beamwidths are measured at 12.1° and 69.1°, respectively. The proposed antenna array’s measured impedance bandwidth ranges from 23.15 to 24.75 GHz, guaranteeing a reflection coefficient (S11) of less than − 10 dB. The suggested antenna's applicability for automotive multi-input multi-output radar has been validated.

Evaluation of imaging dose for the cone beam computed tomography system of the ring-shaped radiotherapy unit using the different International Electrotechnical Commission approaches

PurposeDaily imaging is mandatory for the Halcyon and Ethos, which are the new ring-shaped radiotherapy units. This study quantified the imaging dose of the cone beam computed tomography (CBCT) system of the new ring-shaped radiotherapy units using the latest International Electrotechnical Commission (IEC) measurement approach. Furthermore, we investigated the influence of collimation width on the imaging dose and compared the imaging dose values obtained using three IEC measurement approaches. MethodsThe IEC has proposed three approaches for measuring CTDI100: traditional, transition, and the latest measurement approach. In addition to the dose measured on the CTDI phantom, the latest IEC approach uses correction factors measured in free air for CTDI100 measurement. The imaging dose was measured for all scanning protocols using the latest IEC approach. To investigate the influence of the collimation width on the imaging dose, the associated dose metric values were evaluated and compared using different IEC approaches to increase the collimation width (2–28 cm). Furthermore, the imaging system output stability was evaluated by repeated measurements. ResultsIn the repeated measurements, the standard deviation of CTDIw value for the pelvis and head protocol was 0.05 mGy and 0.01 mGy, respectively. And the coefficient of variation was 0.21% and 0.23%, respectively. The measured CTDIw values of the clinical protocols were higher than the vendor-provided value, except for the pelvis protocol. The value of CTDIw changes little with increasing collimation width using the latest IEC approach. The traditional and transition approaches either underestimated or overestimated the value of CTDIw, as measured by the latest IEC approach, with increasing collimation width. ConclusionsThe new ring-shaped radiotherapy units imaging system has good output stability. The CTDIw values measured for clinical scanning protocols generally exceeded the vendor-provided values. Compared with the latest IEC CTDI100 measurement approach, the traditional and transition measurement approaches were more influenced by beam width. For wide CBCT scans, rationalizing the choice of imaging dose measurement approach is important.

Circular Microstrip Patch Antenna : Design, Simulation, and Analysis For 5G Applications

Microstrip Patch Antennas (MPA) have emerged as a remarkable discovery in the period of miniaturization, despite the fact that advancements in antenna engineering have resulted in the fast growth of communication networks. This work encompasses the design, modelling, and analysis of circular microstrip patch antennas. The resonant frequency of the recommended patch antennas is 9 GHz, which is within the X band range. Ansys HFSS software was used to create them using Rogers RT/duroid 5880 material, which has a dielectric constant of 2.2. Five performance measures were utilized to examine the recommended MPAs: return loss, bandwidth, VSWR, gain, and HPBW. The required information refers to the measurements of return loss, voltage standing wave ratio (VSWR), and half power beamwidth (HPBW). The recommended antennas are ideal for use in radar, wireless, and satellite applications.

Effects of laser shock peening on silicon nitride ceramic with varying sintering additive ratios

Silicon nitride ceramic (Si3N4) for industrial applications is conventionally manufactured with sintering additives, and the properties of Si3N4 change significantly based on the contents of these additives. In this study, we investigate the effects of laser shock peening (LSP) on Si3N4 sintered with varying ratios of sintering additives.The Si3N4 samples were sintered with a mixture of Y2O3, MgO, and SiO2 sintering additives at 5, 7, 9, and 11 wt%. A Nd:YAG laser (wavelength = 532 nm, maximum pulse energy = 1.4 J, repetition rate = 10 Hz, pulse duration = 8 ns, beam diameter = 11 mm, top-hat profile) was used to irradiate the Si3N4 samples. The samples were coated with a protective layer (100 μm thick aluminum foil) and a water layer to confine plasma. LSP of Si3N4 with 5% sintering additives resulted in a slight change in surface hardness but a 77% decrease in surface compressive residual stress. In contrast, LSP of Si3N4 with 11% sintering additives led to a simultaneous increase in surface hardness (7.3%) and surface compressive residual stress (67%), indicating a significant difference in the effectiveness of LSP depending on the ratio of sintering additives. Additionally, the surface of Si3N4 with 11% sintering additives showed evidence of grain refinement after LSP. It was demonstrated that the bending strength of Si3N4 with 11% sintering additives increased by 15.2%, and the depth of the fracture origin was significantly deepened.

The effects of optimized microstructured surfaces on bond strength and durability of NPJ-printed zirconia

ObjectivesThis study was to investigate the effects of optimized microstructured surfaces on bond strength and bond durability of the latest nanoparticle jetting (NPJ)-printed zirconia. MethodsZirconia microstructured surfaces with different geometries and void volume were analyzed through three-dimensional finite element analysis for surface micromorphology optimization. Zirconia disks and cylinders were additively manufactured by an NPJ 3D printer (N = 128). They were randomly divided into four groups based on surface micromorphology optimization and airborne-particle abrasion (APA) treatment before they were bonded using 10-methacryloloxydecyl dihydrogen phosphate (MDP) containing resin cement (Clearfil SA luting cement). The shear bond strengths (SBSs) were tested before and after 10,000 thermocycles and were analyzed by one-way ANOVA analysis. Failure modes were determined by optical microscopy. Zirconia surfaces were analyzed with X-ray diffraction, scanning electron microscopy, and three-dimensional interference microscopy. ResultsThe optimized microstructured surface was characterized by circular microstructures with 60 % void volume, about 20 µm of depths, about 10 µm of undercuts, and consistent beam widths. The optimized microstructured surface combined with APA treatment and MDP-containing resin cement possessed the highest SBSs both before and after thermocycling aging (P＜0.05). The greater reductions of zirconia bond strengths occurred when the zirconia were not treated with APA (P＜0.05). SignificanceThe optimized microstructured zirconia surface with circular microstructures and 60 % void volume fabricated by the latest NPJ printing technology could greatly enhance the zirconia bond strength and durability in combination with APA treatment and application of MDP-containing resin cement, which might be promising for adhesively bonded indirect restorations of NPJ-printed zirconia.

High-precision measurement of trace-level Nb, Sn, Ta, and W in rutile using electron probe microanalysis

The composition of Nb, Sn, Ta, and W in rutile can serve as a highly sensitive indicator for the mineralization of these elements. This information could also be utilized to discriminate different stages of ore deposition or distinct types of deposits. Precise and accurate measurement of trace-level concentrations of these elements in rutile is imperative for such investigations, and can be effectively achieved using electron probe microanalysis (EPMA). In this study, we employed a CAMECA SX5 field emission EPMA to develop an optimal approach for precise and accurate analysis of trace-level Nb, Sn, Ta, and W in rutile. The analytical conditions include an accelerating voltage of 25 kV, a beam current of 200 nA, a beam diameter of 1 μm, optimized background positions for Nb Lα, Sn Lα, Ta Lα, and W Lα, and aggregate intensity counting with a peak counting time of 240 s for each element per spectrometer. The detection limits (3σ) for Nb, Sn, Ta, and W ranged from 22 to 53 ppm. The highest achievable spatial resolution was approximately 4.3 μm. Analytical results obtained from potential rutile standards R10 and R632 were consistent with previous studies within the errors. The precision of these results varied from 1% to 10% (1σ), excluding W in R10 and Nb and Ta in R632, which exceeds the precision achieved in previous EPMA studies. We recommend using R10 as a reference standard for analyses of trace-level Nb, Sn, and Ta, and R632 as an appropriate reference standard for Sn and W when quantifying rutile using EPMA.

Experimental demonstration of a silicon nanophotonic antenna for far-field broadened optical phased arrays

Optical antennas play a pivotal role in interfacing integrated photonic circuits with free-space systems. Designing antennas for optical phased arrays ideally requires achieving compact antenna apertures, wide radiation angles, and high radiation efficiency all at once, which presents a significant challenge. Here, we experimentally demonstrate a novel ultra-compact silicon grating antenna, utilizing subwavelength grating nanostructures arranged in a transversally interleaved topology to control the antenna radiation pattern. Through near-field phase engineering, we increase the antenna’s far-field beam width beyond the Fraunhofer limit for a given aperture size. The antenna incorporates a single-etch grating and a Bragg reflector implemented on a 300-nm-thick silicon-on-insulator (SOI) platform. Experimental characterizations demonstrate a beam width of 44°×52° with −3.22 dB diffraction efficiency, for an aperture size of 3.4 μm×1.78 μm. Furthermore, to the best of our knowledge, a novel topology of a 2D antenna array is demonstrated for the first time, leveraging evanescently coupled architecture to yield a very compact antenna array. We validated the functionality of our antenna design through its integration into this new 2D array topology. Specifically, we demonstrate a small proof-of-concept two-dimensional optical phased array with 2×4 elements and a wide beam steering range of 19.3º × 39.7º. A path towards scalability and larger-scale integration is also demonstrated on the antenna array of 8×20 elements with a transverse beam steering of 31.4º.

Generalized dark hollow sine-Gaussian beam and its propagation properties

A model of the generalized dark hollow sine-Gaussian beam (GDHsGB) is proposed to uniformly describe both conventional dark hollow beams (DHBs) and anomalous dark hollow beams (ADHBs) with circular or elliptic geometrical patterns. Using the Collins formula, we derive the analytical expression for GDHsGBs propagating in ABCD paraxial optical systems. We analyze the evolution of the intensity pattern and beam width of circular ADHBs, as well as the ellipticity of elliptic ADHBs, providing mathematical expressions for these physical quantities. The results reveal various evolution forms based on beam parameters, with elliptic ADHBs exhibiting more intricate propagation behavior compared to circular ADHBs. By controlling parameters, the intensity pattern of elliptic ADHBs undergoes a transformation into a petal-like distribution in the near field, later reverting to its original elliptic configuration but rotated 90° from its initial orientation on the source plane in the far field.

A DDoS Tracking Scheme Utilizing Adaptive Beam Search with Unmanned Aerial Vehicles in Smart Grid

As IoT technology advances, the smart grid (SG) has become crucial to industrial infrastructure. However, SG faces security challenges, particularly from distributed denial of service (DDoS) attacks, due to inadequate security mechanisms for IoT devices. Moreover, the extensive deployment of SG exposes communication links to attacks, potentially disrupting communications and power supply. Link flooding attacks (LFAs) targeting congested backbone links have increasingly become a focal point of DDoS attacks. To address LFAs, we propose integrating unmanned aerial vehicles (UAVs) into the Smart Grid (SG) to offer a three-dimensional defense perspective. This strategy includes enhancing the speed and accuracy of attack path tracking as well as alleviating communication congestion. Therefore, our new DDoS tracking scheme leverages UAV mobility and employs beam search with adaptive beam width to reconstruct attack paths and pinpoint attack sources. This scheme features a threshold iterative update mechanism that refines the threshold each round based on prior results, improving attack path reconstruction accuracy. An adaptive beam width method evaluates the number of abnormal nodes based on the current threshold, enabling precise tracking of multiple attack paths and enhancing scheme automation. Additionally, our path-checking and merging method optimizes path reconstruction by merging overlapping paths and excluding previously searched nodes, thus avoiding redundant searches and infinite loops. Simulation results on the Keysight Ixia platform demonstrate a 98.89% attack path coverage with a minimal error tracking rate of 2.05%. Furthermore, simulations on the NS-3 platform show that drone integration not only bolsters security but also significantly enhances network performance, with communication effectiveness improving by 88.05% and recovering to 82.70% of normal levels under attack conditions.

Pressure scaling of femtosecond laser filamentation in air: Prospects for long-range atmospheric propagation

The principal trend of the last decades is the use of laser systems generating ultrashort (femtosecond) laser pulses of gigawatt and terawatt power in various spheres of human activity, in particular, in atmospheric researches. One of the important challenges within this problem is the use of the unique nonlinear optical phenomena manifesting in the atmosphere as pulse self-focusing, light-induced ionization of the medium, plasma generation and laser filamentation for remote laser diagnostics of the aerosol-gas constituents and efficient delivery of concentrated optical energy over long distances in the atmosphere. This brings to the forefront the problem of increasing the actual length of the nonlinear interaction of an ultrashort laser pulse with medium to kilometer distances whereas real physical length of most existing optical paths is usually limited by several hundred meters. We propose and theoretically substantiate an effective way to solve the problem of long-range atmospheric filamentation by transforming the propagation medium itself using short laboratory traces and optical cuvettes with gases of increased (over-atmospheric) pressure. Following the scaling laws of the optical characteristics of pressurized medium, the coefficients describing the interaction between optical wave and medium increase along with the increase in gas pressure. By the numerical simulations in the framework of the nonlinear Schrodinger equation, we show that this makes it possible to emulate kilometers-long atmospheric traces for laser beams of centimeter diameter and terawatt pulse power on the laboratory scale (tens of meters).

Fundamental precision limits of fluorescence microscopy: a perspective on MINFLUX.

Over the past years, fluorescence microscopy (FM) has steadily progressed in increasing the localization precision of fluorescent emitters in biological samples and led to new claims, whose rigorous validation remains an outstanding problem. We present a novel, to the best of our knowledge, multi-parameter estimation framework that captures the full complexity of a single-emitter FM localization experiment. We showcase our method with Minimum Flux (MINFLUX) microscopy, among the highest-resolution approaches, demonstrating that (i) the localization precision can be increased only by turning the illumination intensity up, thus increasing the risk of photo-bleaching, and it is independent from the beams' separation, and (ii) in presence of background noise, the localization precision decreases with the beams' separation. Finally, we apply our method to Minimum Flux Stimulated Emission Depletion (MINSTED) microscopy, showing that a reduction of the beam width can provide similar performance to MINFLUX.