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Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications The main objective of this book is to present the basic theoretical principles and practical applications for the classical interferometric techniques and the most advanced methods in the field of modern fringe pattern analysis applied to optical metrology. A major novelty of this work is the presentation of a unified theoretical framework based on the Fourier description of phase shifting interferometry using the Frequency Transfer Function (FTF) along with the theory of Stochastic Process for the straightforward analysis and synthesis of phase shifting algorithms with desired properties such as spectral response, detuning and signal-to-noise robustness, harmonic rejection, etc.

With the astrophysics community working toward the first observations and characterizations of Earth-like exoplanets, interest in space-based nulling interferometry has been renewed. This technique promises unique scientific and technical advantages by enabling direct mid-infrared observations. However, concept studies of nulling interferometers often overlook the impact of systematic noise caused by instrument perturbations. Earlier research introduced analytical and numerical models to address instrumental noise; building on these results, we reproduce key simulations and report that the noise in the differential output of nulling interferometers follows a non-Gaussian distribution. The presence of non-Gaussian noise challenges the validity of classical hypothesis tests in detection performance estimates, as their reliance on Gaussian assumptions leads to overconfidence in detection thresholds. For the first time, we derive the true noise distribution of the differential output of a dual Bracewell nulling interferometer, demonstrating that it follows iterative convolutions of Bessel functions. Understanding this noise distribution enables a refined formulation of hypothesis testing in nulling interferometry, leading to a semianalytical prediction of detection performance. This computationally efficient instrument model, implemented in a publicly available codebase, is designed for integration into science yield predictions for nulling interferometry mission concepts. It will play a key role in refining key mission parameters for the Large Interferometer For Exoplanets.

The polarized images of the supermassive black hole Messier 87* (M87*) produced by the Event Horizon Telescope (EHT) provide a direct view of the near-horizon emission from a black hole accretion and jet system. The EHT theoretical analysis of the polarized M87* images compared thousands of snapshots from numerical models with a variety of spins, magnetization states, viewing inclinations, and electron energy distributions, and found a small subset consistent with the observed image. In this article, we examine two models favored by EHT analyses: a magnetically arrested disk with moderate retrograde spin and a magnetically arrested disk with high prograde spin. Both have electron distribution functions that lead to strong depolarization by cold electrons. We ray trace five snapshots from each model at 22, 43, 86, 230, 345, and 690 GHz to forecast future very long baseline interferometry (VLBI) observations and examine limitations in numerical models. We find that even at low frequencies where optical and Faraday rotation depths are large, approximately rotationally symmetric polarization persists, suggesting that shallow depths dominate the polarization signal. However, morphology and spectra suggest that the assumed thermal electron distribution is not adequate to describe emission from the jet. We find 86 GHz images show a ringlike shape determined by a combination of plasma and spacetime imprints, smaller in diameter than recent results from the Global mm-VLBI Array. We find that the photon ring becomes more apparent with increasing frequency, and is more apparent in the retrograde model, leading to large differences between models in asymmetry and polarization structure.

Atomic interferometry measurements of the gravitational force on free-falling atoms provide improved constraints on certain scalar field theories trying to explain dark energy. Traditional gravity measurements use bulk masses to both source and probe gravitational fields 1 . Matter-wave interferometers enable the use of probe masses as small as neutrons 2 , atoms 3 and molecular clusters 4 , but still require fields generated by masses ranging from hundreds of kilograms 5 , 6 to the entire Earth. Shrinking the sources would enable versatile configurations, improve positioning accuracy, enable tests for beyond-standard-model (‘fifth’) forces, and allow observation of non-classical effects of gravity. Here we detect the gravitational force between freely falling caesium atoms and an in-vacuum, miniature (centimetre-sized, 0.19 kg) source mass using atom interferometry. Sensitivity down to gravitational strength forces accesses the natural scale 7 for a wide class of cosmologically motivated scalar field models 8 , 9 of modified gravity and dark energy. We improve the limits on two such models, chameleons 9 and symmetrons 10 , 11 , by over two orders of magnitude. We expect further tests of dark energy theories, and measurements of Newton’s gravitational constant and the gravitational Aharonov–Bohm effect 12 .

Kibble-Zurek theory models the dynamics of spontaneous symmetry breaking, which plays an important role in a wide variety of physical contexts, ranging from cosmology to superconductors. We explored these dynamics in a homogeneous system by thermally quenching an atomic gas with short-range interactions through the Bose-Einstein phase transition. Using homodyne matter-wave interferometry to measure first-order correlation functions, we verified the central quantitative prediction of the Kibble-Zurek theory, namely the homogeneous-system power-law scaling of the coherence length with the quench rate. Moreover, we directly confirmed its underlying hypothesis, the freezing of the correlation length near the transition. Our measurements agree with a beyond-mean-field theory and support the expectation that the dynamical critical exponent for this universality class is z = 3/2.

Very long baseline interferometry observations reveal that relativistic jets like the one in M87 have a limb-brightened, double-edged structure. Analytic and numerical models struggle to reproduce this limb-brightening. We propose a model in which we invoke anisotropy in the distribution function of synchrotron-emitting nonthermal electrons such that electron velocities are preferentially directed parallel to magnetic field lines, as suggested by recent particle-in-cell simulations of electron acceleration and the effects of synchrotron cooling. We assume that the energy injected into nonthermal electrons is proportional to the jet Poynting flux, and we account for synchrotron cooling via a broken power-law energy distribution. We implement our emission model in both general relativistic magnetohydrodynamic (GRMHD) simulations and axisymmetric force-free electrodynamic (GRFFE) jet models and produce simulated jet images at multiple scales and frequencies using polarized general relativistic radiative transfer. We find that the synchrotron emission is concentrated parallel to the local helical magnetic field and that this feature produces limb-brightened jet images on scales ranging from tens of microarcseconds to hundreds of milliarcseconds in M87. We present theoretical predictions for horizon-scale M87 jet images at 230 and 345 GHz that can be tested with next-generation instruments. Due to the scale-invariance of the GRMHD and GRFFE models, our emission prescription can be applied to other targets and serve as a foundation for a unified description of limb-brightened synchrotron images of extragalactic jets.

The 2016 moment magnitude (Mw) 7.8 Kaikoura earthquake was one of the largest ever to hit New Zealand. Hamling et al. show with a new slip model that it was an incredibly complex event. Unlike most earthquakes, multiple faults ruptured to generate the ground shaking. A remarkable 12 faults ruptured overall, with the rupture jumping between faults located up to 15 km away from each other. The earthquake should motivate rethinking of certain seismic hazard models, which do not presently allow for this unusual complex rupture pattern. Science, this issue p. eaam7194 On 14 November 2016 (local time), northeastern South Island of New Zealand was struck by a major moment magnitude (Mw) 7.8 earthquake. The Kaikoura earthquake was the most powerful experienced in the region in more than 150 years. The whole of New Zealand reported shaking, with widespread damage across much of northern South Island and in the capital city, Wellington. The earthquake straddled two distinct seismotectonic domains, breaking multiple faults in the contractional North Canterbury fault zone and the dominantly strike-slip Marlborough fault system. Earthquakes are conceptually thought to occur along a single fault. Although this is often the case, the need to account for multiple segment ruptures challenges seismic hazard assessments and potential maximum earthquake magnitudes. Field observations from many past earthquakes and numerical models suggest that a rupture will halt if it has to step over a distance as small as 5 km to continue on a different fault. The Kaikoura earthquake's complexity defies many conventional assumptions about the degree to which earthquake ruptures are controlled by fault segmentation and provides additional motivation to rethink these issues in seismic hazard models. Field observations, in conjunction with interferometric synthetic aperture radar (InSAR), Global Positioning System (GPS), and seismology data, reveal the Kaikoura earthquake to be one of the most complex earthquakes ever recorded with modern instrumental techniques. The rupture propagated northward for more than 170 km along both mapped and unmapped faults before continuing offshore at the island's northeastern extent. A tsunami of up to 3 m in height was detected at Kaikoura and at three other tide gauges along the east coast of both the North and South Islands. Geodetic and geological field observations reveal surface ruptures along at least 12 major crustal faults and extensive uplift along much of the coastline. Surface displacements measured by GPS and satellite radar data show horizontal offsets of ~6 m. In addition, a fault-bounded block (the Papatea block) was uplifted by up to 8 m and translated south by 4 to 5 m. Modeling suggests that some of the faults slipped by more than 20 m, at depths of 10 to 15 km, with surface slip of ~10 m consistent with field observations of offset roads and fences. Although we can explain most of the deformation by crustal faulting alone, global moment tensors show a larger thrust component, indicating that the earthquake also involved some slip along the southern end of the Hikurangi subduction interface, which lies ~20 km beneath Kaikoura. Including this as a fault source in the inversion suggests that up to 4 m of predominantly reverse slip may have occurred on the subduction zone beneath the crustal faults, contributing ~10 to 30% of the total moment. Although the unusual multifault rupture observed in the Kaikoura earthquake may be partly related to the geometrically complex nature of the faults in this region, this event emphasizes the importance of reevaluating how rupture scenarios are defined for seismic hazard models in plate boundary zones worldwide. (A and B ) Photos showing the coastal uplift of 2 to 3 m associated with the Papatea block [labeled in (C)]. The inset in (A) shows an aerial view of New Zealand. Red lines denote the location of known active faults. The black box indicates the Marlborough fault system. (C ) Three-dimensional displacement field derived from satellite radar data. The vectors represent the horizontal displacements, and the colored background shows the vertical displacements. On 14 November 2016, northeastern South Island of New Zealand was struck by a major moment magnitude (Mw) 7.8 earthquake. Field observations, in conjunction with interferometric synthetic aperture radar, Global Positioning System, and seismology data, reveal this to be one of the most complex earthquakes ever recorded. The rupture propagated northward for more than 170 kilometers along both mapped and unmapped faults before continuing offshore at the island's northeastern extent. Geodetic and field observations reveal surface ruptures along at least 12 major faults, including possible slip along the southern Hikurangi subduction interface; extensive uplift along much of the coastline; and widespread anelastic deformation, including the ~8-meter uplift of a fault-bounded block. This complex earthquake defies many conventional assumptions about the degree to which earthquake ruptures are controlled by fault segmentation and should motivate reevaluation of these issues in seismic hazard models.

It is of great significance to obtain timely and accurate information of surface subsidence caused by mining. The probability integral method (PIM) model is more suitable for mining subsidence prediction in China and has been widely used. However, the PIM model has the question on too fast convergence in predicting at the edge of subsidence basin. In recent years, many scholars have studied a lot of subsidence monitoring methods in the coal mine area using the technical advantages of differential interferometry synthetic aperture radar (D-InSAR). But, serious incoherence of interferometry phase occurs because of the large gradient subsidence of mining area, which leads to the inability to accurately obtain large gradient subsidence of surface. Meanwhile, PIM model is more suitable for static prediction of mining subsidence, and has certain defects in dynamic prediction in the process of mining subsidence. In view of the above shortcomings, the improved PIM (IPIM) prediction model was first introduced through improving the PIM model in the paper, and the IPIM-G dynamic prediction model was constructed based on the PIM model and the Gompertz time function for mine-area mining. Then, a method was used to invert the parameters of the IPIM-G dynamic prediction model using time-series superposition results of surface subsidence monitored by the D-InSAR technology, and then the parameters obtained by inversion were applied to the IPIM-G model for mining subsidence prediction. The model was applied to a coal mine in Huaibei mining area, Anhui Province, its average RMSE is 138 mm and the average RMSE at the edge is 2.8 mm. The accuracy of IPIM-G dynamic prediction model is 88% higher than the monitoring results of the D-InSAR technology in obtaining the gradient subsidence information of the subsidence basin in the mining area. The results show that the model proposed in this paper can provide theoretical support for mining and production planning in the mining area.

Ultrashort optical pulses propagating in a dissipative nonlinear system can interact and bind stably, forming optical soliton molecules. Soliton molecules in ultrafast lasers are under intense research focus and present striking analogies with their matter molecules counterparts. The recent development of real-time spectral measurements allows probing the internal dynamics of an optical soliton molecule, mapping the dynamics of the pulses’ relative separations and phases that constitute the relevant internal degrees of freedom of the molecule. The soliton-pair molecule, which consists of two strongly bound optical solitons, has been the most studied multi-soliton structure. We here demonstrate that two soliton-pair molecules can bind subsequently to form a stable molecular complex and highlight the important differences between the intra-molecular and inter-molecular bonds. The dynamics of the experimentally observed soliton molecular complexes are discussed with the help of fitting models and numerical simulations, showing the universality of these multi-soliton optical patterns. It has recently been shown that optical solitons can form stably bound states, so-called soliton molecules. Here, Wang et al. demonstrate stable soliton molecule complexes and explore the different bonds represented by the inter- and intra-molecular coupling.

In the study of quantum limits to parameter estimation, the high dimensionality of the density operator and that of the unknown parameters have long been two of the most difficult challenges. Here, we propose a theory of quantum semiparametric estimation that can circumvent both challenges and produce simple analytic bounds for a class of problems in which the dimensions are arbitrarily high, few prior assumptions about the density operator are made, but only a finite number of the unknown parameters are of interest. We also relate our bounds to Holevo’s version of the quantum Cramér-Rao bound, so that they can inherit the asymptotic attainability of the latter in many cases of interest. The theory is especially relevant to the estimation of a parameter that can be expressed as a function of the density operator, such as the expectation value of an observable, the fidelity to a pure state, the purity, or the von Neumann entropy. Potential applications include quantum state characterization for many-body systems, optical imaging, and interferometry, where full tomography of the quantum state is often infeasible and only a few select properties of the system are of interest.