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Article
Physical Sciences
Optics and Photonics

Milena Dylko

,

Andrey Novitsky

Abstract: Parity-time (\( \mathcal{PT} \)) symmetry offers a well-established route for enhancing light-matter interaction by means of formation of exceptional points. The model of plane waves is commonly used for finding and investigating exceptional points in open multilayer systems. Here we study behaviors of Bessel light beams in PT-symmetric multilayer structures. We adopt the method of scattering matrices to non-paraxial Bessel beams and determine positions of exceptional points depending on the transverse wave number and non-Hermiticity parameter for both propagating and evanescent beams. We reveal the evolution of exceptional and diabolic lines when the number of layers changes and find their link to the transmission and reflection spectra. This research may pave the avenue for exploiting light beams in non-Hermitian photonics.

Article
Physical Sciences
Optics and Photonics

Soroush Khosravi

,

Joshua Houseman

,

Finley Romano

,

Om Jethwa

,

Rick Trebino

Abstract: We present a novel deep learning-based technique for measuring linear-chirp instability of ultrashort laser pulse trains directly from their Frequency-Resolved Optical Gating (FROG) traces. This approach requires neither an iterative retrieval algorithm nor pre- or post-processing steps, enabling real-time measurements. Although our proof-of-concept convolutional neural networks (CNNs) are moderately deep and trained on compact datasets, they demonstrate resilience to significant additive noise (up to 14%) and high robustness across a wide range of laser pulse parameter variations. When tested on unseen data, our classification network (ICN-1) achieved an accuracy of 80.0% (with an off-by-one accuracy of 100.0%), and our regression network (IRN-2) yielded a normalized root-mean-square error (NRMSE) of 10.2%.

Article
Physical Sciences
Optics and Photonics

Yingchun Li

,

Nattamai Bhuvanesh

,

Tefera Tesema

,

Tony D. Green

,

Jasmin E. Foster

,

Zhenhuan Yi

Abstract: Fulgides are a class of organic compounds that have been extensively investigated because of their photochromic properties and the wide range of applications arising from them. Fulgimides, an important subclass of fulgides, contain a succinimide moiety in place of the succinic anhydride functionality found in fulgides. Isofulgimides are constitutional isomers of fulgimides in which the succinimide ring is replaced by a 5-iminodihydrofuran-2(3H)-one moiety. Although isofulgimides also exhibit photochromism, their synthesis and photochromic behavior have received considerably less attention than those of other fulgide derivatives. Reports on isofulgimides remain scarce, and crystallographic investigations are particularly limited. As part of our ongoing efforts to develop novel photochromic compounds, we synthesized a unique dimeric isofulgimide featuring an N–N single-bond linkage. Herein, we report its synthesis, crystal structure, and photochromic properties.

Review
Physical Sciences
Optics and Photonics

Vladimir L. Kalashnikov

,

Irina T. Sorokina

Abstract: Thermodynamic reasoning has entered nonlinear optics along three routes—wave-turbulence kinetics, the equilibrium thermodynamics of highly multimoded systems, and the statistical mechanics of mode locking—but in each the modal basis or coarse-graining cutoff is specified externally, by a waveguide, a spectral window, or a cavity bandwidth. This review asks what survives when the object is a strongly chirped dissipative soliton of the complex cubic–quintic Ginzburg–Landau equation, a structure that exists only because energy flows through it. The mechanism that localizes such a pulse also bounds its spectrum, separating the correlation function into a graining scale set by the spectral cutoff and a collective scale set by the spectral core. Their ratio is selected by the dynamics rather than imposed, and closed-form adiabatic spectra in both dispersion regimes compress the parameter space onto a single master diagram carrying a shape entropy, an internal energy, and a continuation-dependent energy–entropy slope. Dissipative soliton resonance is read as spectral core narrowing, intrinsic in normal dispersion and conditional in anomalous dispersion. We state precisely where the borrowed vocabulary becomes strained: a deterministic pulse remains first-order coherent, so the scale ratio is not yet a count of statistical degrees of freedom, and the promotion requires an ensemble-coherence calibration that is specified but not assumed. The result is a falsifiable framework for dissipative-soliton coherence, energy scaling, and stability, with experimental signatures and design criteria for chirped-pulse oscillators.

Review
Physical Sciences
Optics and Photonics

Pantelis Mpourazanis

,

Christelle Kielleck

,

Marc Eichhorn

Abstract: Heavy metal fluoride glasses (HMFGs), particularly fluorozirconate glass systems such as ZBLAN have attracted considerable attention due to their unique physical properties, including low phonon energies, wide transparency from the UV to the mid-IR, and high rare-earth ion doping solubility, making them promising materials for photonic applications. This review provides an overview of fluoride glass synthesis methods, structural characteristics, and physical properties of fluorozirconate glasses, with emphasis on glass processing conditions, thermal, mechanical, and optical properties. The structural characteristics are discussed in terms of zirconium–fluorine polyhedral networks and their compositional dependence, while physical properties are analyzed, including glass transition behavior, crystallization tendency, elastic moduli, and infrared transmission. Rare-earth doped Er3+, Ho3+, and Tm3+ ZBLAN glasses are also discussed, which exhibit efficient emissions in the near and mid-IR spectral regions. Although significant progress has been achieved, limitations related to thermal stability, mechanical strength, and incomplete understanding of structure–property relationships persist. Future research should therefore focus on compositional optimization and predictive structural modeling to enable the design of improved fluoride glasses for various applications.

Article
Physical Sciences
Optics and Photonics

Daniel A Nolan

Abstract: We simulate the propagation of a single photon propagating within a modal group, traversing through a multimode optical fiber in the presence of mode coupling. We illustrate the propagation graphically on a group of higher order Poincare spheres. The spheres display the propagation of the light within the modal group including polarization in time and in distance as the transmission proceeds. Thus, the amplitudes and the relative phases within the group can be visualized throughout the transmission, which is novel and very useful to understand the propagation. At the fiber output we show how to recover the input of a classical state using the simulated propagation information displayed on such multiple spheres. Once a classical - modal path is established from end to end, one can transmit quantum states, for example a spatial – time binned QKD (quantum key distribution) code. A quantum state follows this classical path. This can include binary or qudit information. Accessing a photon output state is important for many quantum network applications including quantum key distribution, routing and entanglement swapping. In addition, one can use other modal groups within the same fiber to multiplex other quantum channels. Further, one can multiplex within a modal group using principal modes, a more complicated communication method. Using the spheres, this situation is also discussed. Applications include higher dimensional quantum communications, quantum cryptography, and quantum networks. It is important to point out that in today’s commercial quantum communications systems, the quantum states are single photon states but not entangled. This includes both discrete and continuous variable quantum key distribution systems. In this report we are also looking at single photon systems but not entangled, unless stated otherwise.

Article
Physical Sciences
Optics and Photonics

Yue He

,

Xin Zhang

,

Yunhai Tang

,

Baohua Chen

Abstract: In laser phase transformation hardening and laser cladding processes, the inherent thermodynamic limitations of conventional intensity distributions severely restrict the uniformity of metallurgical reaction. Although programmable beam shaping devices offer flexibility in profile reconstruction, their transmissive structure results in a low laser-induced damage threshold (LIDT), making them unsuitable for long-term stable operation at kilowatt-level power. This study proposes a reflective freeform mirror design method based on the principle of zonal energy mapping. The method constructs energy mapping relations that correlate the irradiance distribution in every sub-region of the incident Gaussian beam with the desired M-shaped irradiance profile. Relying on such mappings, the local surface generatrices of all sub-regions are solved separately; these generatrices are subsequently assembled to yield an integrated mirror surface. Optical performance was verified through Zemax non-sequential ray tracing simulations. The mirror was precisely machined using single-point diamond turning (SPDT). The experimental results show that the measured intensity distribution on the observation screen exhibits a typical M-shaped profile, with a peak-to-valley ratio of 1.28, which is in good agreement with the design target. This method provides a technically feasible and engineering-robust solution for complex thermal flux control requirements in high-power laser material processing.

Review
Physical Sciences
Optics and Photonics

Changle Meng

Abstract: Optical tweezers (OTs) have transformed the ability to manipulate microscopic and nanoscopic matter using light, establishing a non-contact, high-precision platform for probing forces at the picoNewton scale. Since their first realization in the 1980s, OTs have expanded from classical single-beam traps into a diverse set of architectures—including holographic, plasmonic, fiber-based, and integrated photonic platforms—each extending trapping performance across spatial scales and material systems. This review provides a roadmap that integrates fundamental mechanisms, technological advances, and emerging applications of optical tweezers. We begin with a comprehensive analysis of the optical force landscape, encompassing gradient and scattering forces, Brownian fluctuations, viscous drag, hydrodynamic interactions, and quantum electrodynamic effects such as Casimir–van der Waals interactions. We then discuss experimental techniques for trap calibration and potential reconstruction, including equipartition, power spectral density analysis, Stokes drag, and interferometric detection, which together enable force quantification with sub-piconewton precision. Architecturally, we classify OTs into single-beam, holographic, fiber-based, plasmonic, and hybrid photonic systems, highlighting how recent advances in metasurfaces, nanophotonic resonators, and AI-assisted control are reshaping optical trapping into compact, intelligent, and scalable platforms. Applications are reviewed across biology, soft matter, and quantum science—from single-molecule mechanics and cellular biomechanics to nanoparticle assembly and quantum simulations. Finally, we outline current limitations—including diffraction-limited confinement, nanoscale trapping efficiency, and photothermal damage—and discuss future directions toward adaptive beam shaping, cryogenic trapping, and quantum-enhanced tweezers. By bridging theoretical foundations, experimental methodologies, and cross-disciplinary applications, this roadmap aims to provide both a technical reference and a forward-looking vision for the next generation of optical trapping technologies. By synthesizing the theoretical foundations, experimental methodologies, and future-oriented integrations, this review aims to serve as both a technical guide and a visionary roadmap for advancing the next generation of optical trapping platforms.

Article
Physical Sciences
Optics and Photonics

Jihad Zallat

,

Yoshitate Takakura

,

Christian Heinrich

,

Romain Attal

,

Laurent Schwartz

Abstract: The degree of polarization is usually obtained from the coherency matrix or, equivalently, from the mean Stokes vector of a partially polarized optical field. Here, we adopt a complementary geometric viewpoint by representing normalized local Stokes vectors as random directions on the Poincaré sphere. When these directions are described by an effective unimodal von Mises–Fisher distribution, the concentration parameter gives a direct one-to-one description of the degree of polarization through the mean resultant length. This formulation does not define a new independent polarization observable. Instead, it gives the degree of polarization a rotation-invariant information-theoretic meaning, expressed in terms of directional concentration and angular disorder. Within this framework, we derive closed-form expressions for the differential entropy of the von Mises–Fisher distribution and for the Kullback–Leibler divergence between two directional polarization states. The symmetrized divergence further incorporates both differences in concentration and relative orientation on the Poincaré sphere. We also discuss the assumptions, range of validity, and limitations of the single-vMF model, particularly in relation to Gaussian-field statistics and to more general directional models needed for anisotropic or multimodal polarization fluctuations. Overall, this formalism establishes a model-based theoretical framework for entropy and divergence descriptors of unimodal directional polarization and suggests natural extensions toward mixtures of vMF, Bingham, or Kent distributions.

Article
Physical Sciences
Optics and Photonics

T.E Girish

,

P.E Eapen

Abstract: The physical conditions under which the outstanding students of CV Raman : KR Ramanathan in 1923 and KS Krishnan in 1924 could detect Raman scattering in pure liquids ( understood as weak fluorescence in those days) are investigated in detail in this paper. The instrumentation used for these findings is simple consisting of complimentary color filters , sunlight and polarisation detectors. The relationships between Raman shifts, favourable Raman bands in samples and the choice of the complimentary color filters is explained. Raman bands in the favourable region ( 2800-3500 cm-1) existed in the 17 liquids for which is weak fluorescence is detected by Krishnan included water and ethanol for which similar results are found by Ramanathan earlier. We will also discuss the advantages and limitations in the the first report of Raman scattering by Raman and Krishnan to Nature on February 16th in 1928.

Article
Physical Sciences
Optics and Photonics

Yihan Li

,

Joel Y. Y. Loh

Abstract: Short-wave infrared photodetector stacks based on Lead Sulphide (PbS) promise inexpensive and tunable responsivity in the infrared regime. However, the low charge carrier mobilities of PbS require additional electron (ETL) and hole transport (HTL) layers, which modifies the optical absorption Fabry-Perot profile within the two mirror-like electrodes. This creates a coupled design problem, where changes in transport layer thickness can redistribute optical energy away from the absorber or increase loss in the electrodes and transport layers, thus reducing useful absorption. Here, we introduce a cavity-partitioning perspective for planar PbS photodiode stack, treating the air/ITO/ETL/n-PbS/p-PbS/HTL/Au stack as a coupled lossy Fabry-Perot cavity whereby the electron and hole transport layer thicknesses control the partition of absorption between useful PbS photogeneration and parasitic loss. Subsequently, the HTL behaves as a phase tuning layer that can move the field antinode closer to the hole transport layer, while ETL behaves as a field strength control layer that modulates field amplitude near the electron transport layer.

Article
Physical Sciences
Optics and Photonics

Haoran Mu

,

Hsin-Hui Huang

,

Tomas Katkus

,

Nguyen Hoai An Le

,

Jurga Juodkazytė

,

Yoshiaki Nishijima

,

Saulius Juodkazis

Abstract: In femtosecond-laser processing of titania in water, light can induce reduction and oxidation simultane-ously. We follow this duality, in this perspective, from colloidal titania synthesis to hot-electron devices. Femtosecond ablation/fragmentation of an aqueous anatase suspension (515 nm, 230 fs, 5 µJ, fluence F ≈ 25.5 J cm−2/pulse at clamped intensity ∼1013 W cm−2) yields surface-reduced, Ti3+-rich bluish TiO2 – x, while the same optical breakdown generates reactive oxygen species (ROS), among them H2O2 and HO• radicals, which compete by re-oxidising Ti3+. When the reduced titania is decorated with plasmonic nanoparticles (e.g., Au), an n-type plasmonic photo-electrode is realised: sp hot electrons are injected over the Schottky barrier, while the deep d-band supplies oxidising holes. Water oxidation proceeds in stages at potentials well above the formal 1.23 V via the two-electron peroxide route (∼1.77 V) or, for sufficiently energetic holes, via the one-electron HO• route (∼2.7 V). In a biased cell, H2 evolves on Pt through the adsorbed (H2+)ad intermediate. The same Au/semiconductor physics on silicon enables sub-band-gap hot-electron photo-detection. Energy-level diagrams (flat-band and in-contact) and the sp- vs. d-band origin of the injected carriers are discussed.

Article
Physical Sciences
Optics and Photonics

Pierre Jeunesse

,

Yanis Abdedou

,

Mirza Barlas

,

Aloïs Baudry

,

Sylvie Lebrun

Abstract: Optical microfibers are fabricated by pulling classical silica fibers until reaching diameters of a few micrometers or less. These devices are significantly exploited in many science and engineering fields, ranging from fundamental research to practical applications. Despite their many attractive advantages, a major technological challenge remains: heating caused by laser absorption from surface defects and contaminants. In the present study, we propose, for the first time to our knowledge, a novel method to measure the temperature evolution of laser self-heated microfibers in air at a wavelength of 1.48 µm. This method, simple and fast, enables us to investigate the influence of the diameters and lengths of the microfibers. We found that the temperature of the microfibers increases linearly with the power and measured a rise of 70°C for a 1 µm diameter and 20 mm length microfiber at a moderate power of 160 mW. A numerical model considering the microscale and the heat exchange with air is proposed and is adjusted with experimental data, providing values for the thermal transfer coefficient. By investigating power scaling, this work enables the prediction of temperature increases in self-heated microfibers in air, paving the way for new insights into the self-cleaning of microfiber-based devices and for optimized control of light propagation at high power levels.

Article
Physical Sciences
Optics and Photonics

Ranbir Kaur

,

Mohanad Alkaales

,

Mohamed Benyoucef

Abstract: We demonstrate the integration of molecular beam epitaxy (MBE)-grown InP-based quantum dots (QDs) on gold thin films, achieving a fivefold enhancement of emission at telecom wavelengths compared with QDs grown on distributed Bragg reflectors (DBRs). Micro-photoluminescence (µ-PL) spectroscopy reveals a pronounced enhancement of the PL intensity, which is attributed to plasmon-assisted light–matter interaction at the metal–semiconductor interface. Reflectivity measurements reveal a characteristic dip near the QD emission wavelength, consistent with the excitation of surface plasmon resonances (SPRs), indicating favorable spectral overlap between the plasmonic modes and the QD emission. Power-dependent measurements reveal background-free exciton and biexciton emission from single QDs with resolution-limited linewidths. Polarization-dependent measurements demonstrate an ultra-small excitonic fine-structure splitting as low as ~2 μeV. Finally, a statistical analysis of multiple QDs confirms the reproducibility of the observed optical properties.

Article
Physical Sciences
Optics and Photonics

Jianming Wen

Abstract: Time-reversed Young (TRY) interferometry reconstructs interference from a fixed detector by reading out a programmable source-label distribution. We formulate this architecture as a source-coded Bayesian response sensor. For a perturbation parameter θ, the detected source-label histogram is a posterior distribution determined by a programmed source prior, an optical likelihood for a fixed-detector click, and an evidence factor equal to the click probability. The key point is not the Bayesian identity itself, but its physical implementation: in TRY the prior is imposed before propagation and can therefore reshape the response ensemble actually sampled by the detector. We show that the normalized posterior responds through a centered likelihood score, and that the detected-event Fisher information is the posterior variance of this score. This identifies posterior-weighted score contrast, rather than local response magnitude alone, as the relevant sensing resource. The framework separates posterior-shape information from evidence information, giving a resource-aware way to judge near-null response enhancement. It also yields practical design rules: a two-label source code converts a weak perturbation into a fixed-detector label imbalance, while the multiparameter score covariance provides a route to nuisance rejection and gives a minimal-label rank condition for sensing multiple perturbations. A passive double-slit implementation with weak one-slit phase and loss perturbations is proposed, requiring only fixed-detector source scans before and after calibrated perturbations. The results position TRY as a source-programmable Bayesian sensing architecture complementary to conventional detector-plane Young interferometry.

Article
Physical Sciences
Optics and Photonics

Myroslav Strynadko

Abstract: Vortex optical states carrying orbital angular momentum provide an additional degree of freedom for fiber photonic systems [1-3], but their practical use requires efficient and reproducible excitation of guided modes with compatible transverse field structures. Ring-core and multi-ring-core fibers are natural platforms for such states because their annular guiding geometry can support modes with azimuthal phase variation [4-13]. However, a freely generated vortex beam does not automatically provide selective excitation of a prescribed guided mode. In this work, we develop a metasurface-assisted modal-matching framework for coupling vortex optical states into ring-core and multi-ring-core fiber architectures. The metasurface is treated not only as a vortex generator, but as a compact input interface that forms an amplitude–phase–polarization field profile optimized for a target vortex-compatible guided mode. A scalar modal-overlap model is used to evaluate target-mode coupling efficiency, modal purity, OAM-state purity, crosstalk, insertion loss, and tolerance sensitivity. The simulations show that a simple vortex field with the correct azimuthal order provides perfect OAM-state purity in the scalar basis, but only 0.616 target-mode coupling efficiency and 2.10 dB insertion loss. After radial modal matching, the coupling efficiency increases to 0.975 and the insertion loss decreases to 0.11 dB, while other-ring leakage is strongly suppressed. Practical non-idealities are also quantified: eight-level phase quantization still gives 0.950 coupling efficiency, whereas random phase errors with standard deviation reduce it to 0.856. The tolerance analysis identifies lateral displacement, beam-waist mismatch, ring-radius mismatch, phase noise, and polarization mismatch as the dominant limitations. The proposed framework provides a quantitative basis for designing structured-light input interfaces for selective excitation of vortex-compatible modes in annular fiber architectures.

Article
Physical Sciences
Optics and Photonics

Alisher Rajabov

,

Obid Sabirov

,

Bogibek Urinov

,

Gaetano Assanto

,

Usman Sapaev

Abstract: We numerically investigate high-efficiency third-harmonic generation (THG) via cascaded 2 χ(2) processes in orientation-patterned GaAs (OP-GaAs) comprising two quasi-phase- 3 matched sections. The first section, with period Λ1 ≈ 212 μm, is phase-matched for second- 4 harmonic generation (ω → 2ω); the second, with period Λ2 ≈ 182 μm, for sum-frequency 5 generation (ω + 2ω → 3ω). Pumped by a continuous-wave CO2 laser at λ = 10.61 μm, 6 the structure converts the fundamental to the third harmonic at λ3ω ≈ 3.54 μm, within 7 the atmospheric transmission window. By solving the coupled-wave equations with 8 temperature-dependent Sellmeier dispersion for GaAs, we optimize the number of domains 9 in each section and predict THG conversion efficiencies exceeding 90% at a pump intensity 10 of ∼ 100 MW/cm2 in a crystal of total length ≈ 1.6 cm (N1 ≈ 66 domains in the SHG 11 section and N2 ≈ 100 in the SFG section). We also analyze the thermal tuning of the 12 quasi-phase-matching conditions over the range 22–300C, and demonstrate robustness 13 against fabrication tolerances of ±10 domains and ±2% in domain length. Our results 14 establish two-section OP-GaAs as a practical, high-efficiency platform for mid-infrared 15 frequency tripling.

Article
Physical Sciences
Optics and Photonics

Youjun Ma

,

Yongqiang Li

,

Cheng Ju

,

Changhong Li

Abstract: One of the bottlenecks in realizing all-optical computing is the lack of on-chip all-optical logic devices that combine compact, low-loss, and highly robustness. Valley photonic crystals (VPCs) have become an important solution for realizing such devices, relying on the excellent transmission characteristics of topological valley states. However, existing structures still face issues such as limited design flexibility. In this paper, a high-performance topological all-optical logic device based on VPCs consisting of circular ring dielectric columns is designed and demonstrated. By introducing the inner radius as an independent design parameter, we construct a new type of VPC and systematically investigate its influence on the photonic band gap. Based on this, we design a beam splitter with high operational bandwidth and low insertion loss (<0.5 dB), and then realize fundamental OR and XOR logic gates, achieving extinction ratios of 18.9  dB for the OR gate and up to 44  dB for the XOR gate at an operating frequency of 193.5  THz. The platform also supports the NOT gate and, through cascading, can implement more logic functions such as AND gate.

Article
Physical Sciences
Optics and Photonics

Hanyi Zhang

,

Rong Fan

,

Yinzhou Zhi

,

Lulu Fang

,

Wenxuan Cheng

,

Yujie Wang

,

Jianfeng Bao

,

Lijing Li

Abstract: In this study, we present a thermal-aware design of a compact hybrid plasmonic grating (HPG) TE-pass polarizer on X-cut lithium-niobate-on-insulator (LNOI) for fiber-optic gyroscopes (FOGs). In a three-dimensional simulation, the optimization of the trapezoidal sidewall angle (θ = 78°) and the thickness of the Ag grating (13 nm) yield a polarization extinction ratio of 36.2 dB at 1550 nm (with a peak of 41.4 dB at 1548 nm) within a sub-10 μm grating length. This represents a ~3–8 dB improvement over prior LNOI HPG polarizers at the same footprint. A multiphysics thermo-optic analysis over the wide industrial FOG envelope (from −45 to +85°C) demonstrates that the operating-wavelength polarization extinction ratio remains within the range of 24.7–36.2 dB across the entire 130 K span (worst case 24.7 dB at −25°C), constrained solely by a modest 10 pm/°C Bragg detuning stemming from the pronounced (~5) thermo-optic anisotropy of LN. The insertion loss exhibits a negligible drift of merely 0.73 dB. A fabrication-tolerance study identified the Ag thickness as the predominant budgetary constraint (±1 nm tolerance, PER dropping ~10 dB at the resonance edge), while the ridge width and oxide buffer demonstrated comparatively greater flexibility. The device, therefore, fulfills the criteria for FOG-grade polarization suppression across the majority of the operational temperature range. The −25 °C point is established at the 25 dB threshold, thereby providing concrete design guidelines for ensuring environmentally stable on-chip polarization control on LNOI.

Article
Physical Sciences
Optics and Photonics

Cinthia Guadalupe Mata Ramirez

,

Alexander N. Pisarchik

,

Guillermo Huerta Cuellar

,

Juan Hugo García López

,

Samuel Mardoqueo Afanador Delgado

,

Waqar Hussain Shah

,

Rider Jaimes Reátegui

Abstract: The dynamics and synchronization properties of a network of three erbium-doped fiber lasers coupled via higher-order interactions (HOI) are investigated using simplicial complexes. Three coupling configurations are analyzed: first-order only, second-order only, and their combination. Several synchronization metrics, including the average synchronization error, similarity index, and correlation coefficient, are employed to characterize the effects of HOI. The results demonstrate that incorporating second-order coupling substantially enriches the system’s dynamical behavior, revealing a greater diversity of collective states than first-order coupling alone. While complete synchronization is not achieved within the bounded coupling strengths considered, both phase synchronization and anticipation synchronization emerge within distinct parameter intervals.

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