Van Cittert Zernike Research Articles

A model-experiment validation of near-range phase center breakdown

Spatial coherence of a scattered field has helped to improve sensor navigation techniques and field imagery. Under the phase center approximation (PCA), a bistatic sensor geometry is approximated as a monostatic phase center. In the far field, the approximation is valid, and signals with common phase centers are highly coherent. The approximation fails when a non-trivial spatiotemporal delay appears in the sensor’s near-field, and signals that share phase centers are less coherent. A model by Brown and Blanford shows this near-field degradation of PCA, using comparisons between the van Cittert-Zernike Theorem and a point-based scattering model. Experimental validation of the model was performed using an in-air sonar mounted on a linear actuator. The sonar collected returns from a shallow bed filled with plastic beads, which approximates rough interfaces typical of underwater environments. Multiple trials were conducted using a variety of transmitters, receiver placements, and ping intervals. These geometries were simulated using the point-based scattering model and the two sets of data were compared in order to validate the model. Results from this model-data comparison, along with future applications for sonar imaging and navigation, will be discussed.

Optimal transmit apodization for the maximization of lag-one coherence with applications to aberration delay estimation

Phase aberration is one of the major sources of image degradation in medical ultrasound imaging. One of the earliest and simplest techniques to correct for phase aberration involves nearest-neighbor cross correlation to estimate delays between neighboring receive channels and the compensation of aberration delays in a delay-and-sum beamformer. The main challenge is that neighboring receive channels may not have sufficient signal correlation to accurately estimate the aberration delays. Although algorithms such as the translating transmit aperture or the common midpoint gather are designed to perfectly maximize signal correlations between received signals, these algorithms require the use of different transmit apertures for each received signal. Instead, this work proposes the use of a single globally-applicable transmit apodization function that optimizes the lag-one coherence based on the van Cittert–Zernike theorem. For the application to phase aberration correction, it is shown across 20 different zero-mean Gaussian-random aberrators that the proposed optimal apodization function reduces the estimation error in the aberration delay profile from 22.85% to 15.72%.

Autocorrelation functions and power spectral densities of the Stokes parameters in a polarization speckle pattern.

The concept of ensemble-average polarization and coherence has been applied to studying fluctuating Stokes parameters in a polarization speckle observed when coherent light is passed through a birefringent polarization scrambler. With the aid of the ensemble-average van Cittert-Zernike theorem for the propagation of ensemble-average polar-coherence, we invesitgate the autocorrelation functions and power spectra of the Stokes parameters to expose the dependence of the polarization-related scale-size distributions on the optical geometries in which the polarization speckle arises. A generalized concept of the Stokes ensemble-average coherence areas is introduced to deal with the polarization-related average areas associated with polarization speckle.

Effect of micro-range transmission on the imaging sharpness of near-infrared upconversion thin-film silicon-based detectors

Near-infrared photons are converted to visible photons by nonlinear crystal upconversion materials and detected directly by silicon-based detectors, which is an attractive method for near-infrared detection. In this study, the influence of the film thickness of the upconversion material on the imaging sharpness based on the Van Cittert–Zernike theorem is analyzed. The relationship between imaging sharpness of silicon-based detectors and different thicknesses of near-infrared upconversion films is obtained by experimental measurements. The results show that the crystal luminescence area As of the microstructure unit of the upconversion material and the wavelength λ¯ of the converted visible light center remain unchanged with the change of thickness. The crosstalk distance a of the imaging sharpness is proportional to the thickness z of the upconversion material film, which is consistent with the theoretical results.

Coulomb Interactions and the Spatial Coherence of Femtosecond Nanometric Electron Pulses

The transverse coherence of electrons is of utmost importance in high resolution electron microscopes, point-projection microscopes, low-energy electron microscopy, and various other applications. Pulsed versions of many of these have recently been realized, mostly relying on femtosecond laser-triggering electron emission from a sharp needle source. We here observe electron interference fringes and measure how the interference visibility becomes reduced as we increase the emitted electron bunch charge. Due to the extremely strong spatiotemporal confinement of the electrons generated here, we observe the visibility reduction already at average electron bunch charges of less than 1 electron per pulse, owing to the stochastic nature of the emission process. We can fully and quantitatively explain the loss of coherence based on model simulations. Via the van Cittert–Zernike theorem, we connect the visibility reduction to an increase in the effective source size. We conclude by discussing emittance, brightness, and quantum degeneracy, which have direct ramifications to many setups and devices relying on pulsed coherent electrons.

Sensitivity of spatial coherence of sub-bottom scattering to sediment attenuation

The spatial coherence of seafloor scattering is determined by a complex interaction between parameters describing the environment, the sensor, and the measurement geometry. Forward modeling shows that near normal incidence, the spatial coherence of the field backscattered from an isotropic sub-bottom may be sensitive to the sediment attenuation coefficient. This presentation will describe a parametric study investigating the sensitivity of spatial coherence to specific combinations of environment, sensor, and measurement geometry. Particular emphasis will be placed on identifying configurations where the spatial coherence is sensitive to the sediment attenuation coefficient. To support this analysis, a numerical model using the van Cittert–Zernike Theorem and a point-based scattering model will be used to provide forward modeling of the spatial coherence. These models are compared for several simulated environments of increasing complexity to explore the sensitivity of the spatial coherence to sediment attenuation. Agreement between these models will provide a path to developing the collection geometry and algorithms necessary for inversion using simulated data.

Applicability of the Van Cittert-Zernike theorem in a Ronchi shearing interferometer.

Ronchi shearing interferometry is a promising technique for in situ wavefront aberration measurement of the projection lens in advanced photolithography systems. The Van Cittert-Zernike theorem has been used to analyze the interference signal of a Ronchi shearing interferometer in the effective interference area (overlapping area of the ±1st diffraction orders produced by the image grating). However, the applicability of this theorem has not been systematically studied. In this work, the analytical expression of the Ronchigram in this area is derived based on the theory of scalar diffraction and incoherent imaging. The results show that only the object and image grating with infinite diffraction orders can fully satisfy the Van Cittert-Zernike theorem. In the finite diffraction orders case, the theorem can be considered approximately applicable in the overlapping area of the ±3rd diffraction orders produced by the image grating. The applicable area extends to the overlapping area of the ±2nd diffraction orders under a shear ratio of less than 1%, which accounts for 97% of the effective interference area. The theoretical analysis has been verified by simulation and fundamental experiments.

Coherence-Induced Bias Reduction in Synthetic Aperture Sonar Along-Track Micronavigation

Subwavelength motion estimation is vital for the production of focused synthetic aperture sonar (SAS) imagery. The required precision is obtainable from the sonar data itself through a process termed micronavigation. Along-track micronavigation is achieved by a similar technique to that used in correlation velocity logs (CVLs), where sparse estimates of the spatial coherence function are interpolated to estimate the location of the peak coherence and hence estimate the interping vehicle motion. However, along-track micronavigation estimates made using this technique are biased, which limits the utility of these measurements for long-term navigation of autonomous underwater vehicles (AUVs). Three sources of along-track motion estimation bias are considered in this article. First, imperfect temporal registration between the signals results in coherence estimates that are negatively biased as a function of the temporal offset. Second, the sparse estimates of the spatial coherence function are obtained by cross-correlation of complex baseband signals, a process which is known to result in positively biased coherence estimates, especially when the true coherence is low. Finally, mismatches between the underlying spatial coherence function and the interpolation kernel used to estimate the peak coherence location also result in along-track micronavigation bias. In this article, we describe and evaluate three methods for reducing along-track micronavigation bias. We introduce a temporal registration of the signals before coherence estimation, which reduces the impact of negative coherence bias due to temporal offsets. The remaining coherence estimation bias is reduced by combining multiple coherence estimates in a Bayesian coherence estimator. Additionally, an improved interpolation kernel is derived with a significantly improved fit compared to the current gold standard Gaussian interpolation kernel. The improvements in along-track micronavigation accuracy are demonstrated using two simulated data sets, which both allow comparison with ground truth. The first involves direct simulation of the spatial coherence from a given interping geometry using the pulse-echo formulation of the van Cittert–Zernike theorem, while the second involves simulation of raw sonar echo data using a point-scatterer model. Using these simulations, a reduction in along-track micronavigation bias of 48.5%–99.5% is demonstrated, with reductions in along-track micronavigation error standard deviation of up to 34%. This improvement expands the potential for SAS-equipped AUVs to reduce their long-term navigation drift, facilitating longer underwater transits, improved target localization, and reduced track misalignment in repeat-pass operations.

Photonics-Based Simultaneous Angle of Arrival and Frequency Measurement System With Multiple-Target Detection Capability

A photonics-based system for simultaneous angle of arrival (AOA) and frequency measurement is proposed and demonstrated. In the proposed system, an acousto-optic modulator (AOM) based optical frequency shift loop (OFSL) is used to generate an optical frequency-stepped local oscillator (LO). After balanced in-phase and quadrature (I/Q) down-conversion with the frequency-stepped LO, the frequency to be measured is obtained. Thanks to the balanced I/Q down-conversion, the image frequencies and undesired frequency-mixing components can be suppressed, making the system have better adaptability to multiple-target scenarios. In addition, the AOA from different RF sources can also be estimated based on the Van Cittert–Zernike theorem, even if the working bands of these targets overlap with each other. In the proof-of-concept experiment, the measurement of multiple targets with different frequencies is achieved by frequency detection with 0.179 MHz resolution. Moreover, incoherent microwave sources at the same frequency band can also be distinguished in the angular direction, achieving the capacity of simultaneous frequency and AOA measurement of multiple-target.

Vortex particles in axially symmetric fields and applications of the quantum Busch theorem

The possibilities to accelerate vortex electrons with orbital angular momentum (OAM) to relativistic energies and to produce vortex ions, protons, and other charged particles crucially depend on whether the OAM is conserved during the acceleration and on how phase space of the wave packet evolves. We show that both the OAM and a mean emittance of the packet, the latter obeying the Schrödinger uncertainty relation, are conserved in axially symmetric fields of electric and magnetic lenses, typical for accelerators and electron microscopes, as well as in Penning traps. Moreover, a linear approximation of weakly inhomogeneous fields works much better for single packets than for classical beams. We analyze quantum dynamics of the packet’s rms radius ⟨ρ 2⟩, relate this dynamics to a generalized form of the van Cittert–Zernike theorem, applicable at arbitrary distances from a source and for non-Gaussian packets, and adapt the Courant–Snyder formalism to describe the evolution of the packet’s phase space. The vortex beams can therefore be accelerated, focused, steered, trapped, and even stored in azimuthally symmetric fields and traps, somewhat analogously to the classical angular-momentum-dominated beams. We also give a quantum version of the Busch theorem, which states how one can produce vortex electrons with a magnetized cathode during either field- or photoemission, as well as vortex ions and protons by using a magnetized stripping foil employed to change a charge state of ions. Spatial coherence of the packets plays a crucial role in these applications and we provide the necessary estimates for particles of different masses.

Spatial coherence characterization of light: An experimental study using digital micromirror devices

We present spatial coherence measurements of partially coherent light in the far-field of incoherent sources with an experimental setup based on the Thompson–Wolf and Partanen–Turunen–Tervo experiments, to be performed in the context of a possible solar coherence measurement space instrument. The optical setup consists of a telescope to collimate light from a source, to diffract it by a digital micromirror device implementing a Young double-aperture interferometer in retroreflection mode, and finally to image the source into a two-dimensional sensor. Two multimode optical fibers with different diameters were used as incoherent sources and the results obtained for the spectral degree of coherence are compared to those expected from the van Cittert–Zernike theorem.

Entanglement Enabled Intensity Interferometry of different wavelengths of light

We propose methods to perform intensity interferometry of photons having two different wavelengths. Distinguishable particles typically cannot interfere with each other, but we overcome that obstacle by processing the particles via entanglement and projection so that they lead to the same final state at the detection apparatus. Specifically, we discuss how quasi-phase-matched nonlinear crystals can be used to convert a quantum superposition of light of different wavelengths onto a common wavelength, while preserving the phase information essential for their meaningful interference. We thereby gain access to a host of new observables, which can probe subtle frequency correlations and entanglement. Further, we generalize the van Cittert–Zernike formula for the intensity interferometry of extended sources, demonstrate how our proposal supports enhanced resolution of sources with different spectral character, and suggest potential applications.

Complex degree of spatial coherence over a plane irradiated by quasi-monochromatic light sources

We revisit the concept of optical spatial coherence by considering the case of the complex degree of spatial coherence induced over a plane by a quasi-monochromatic irradiating source. We apply the Van Cittert–Zernike theorem and exploit a Lloyd’s two-beam interferometer setup with its two small coherent light sources, as a way to illustrate the calculation of such complex spatial coherence, and because such setup offers a simple way to vary the separation d between its two irradiating sources, which then allows one to analytically obtain the functional dependence of the complex spatial coherence upon that separation d. We then consider the more general case in which the two small light sources of the experimental setup are spatially incoherent and again obtain the complex degree of coherence over a plane. This provides a simple, yet rigorous way, to present the subject of spatial coherences and field correlations to undergraduate physics students. An interesting link with astronomical interferometry, when images of, say, binary stellar objects are acquired, is established.

Low frequency synthetic aperture sonar: Spatial coherence analyse

Spatial coherence is a widely studied notion for HF SAS. It allows the estimation of the navigation of the platform with an accuracy well below the wavelength of the system . It also allows, for high-frequency SAS images, the detection of stable reflectors generally symptomatic of manufactured objects. The theoretical background of spatial coherence is based on a derivation of the Van Cittert Zernike (VCZ) optics theorem adapted to acoustic. However, this theorem uses some assumptions (narrow band signals, Fresnel approximation, Born approximation and homogeneity of the scattering media) that are not necessary met for low frequency system. In this paper effects of wideband systems and presence of a target in the insonified volume are investigated. Do to so, the study is based on a simulation tool whose results are compared with real data from trials conducted by CMRE with a wideband LFSAS. Thanks to the use of a 2-D array, the two dimensional coherence can be analyzed. A sub-band study enables to respect the narrowband VCZ assumption. A statistical study on distribution of degree of coherence estimates allows to point out the presence of strong scatterers symptomatic of manufactured objects.

Measurement of the full complex degree of coherence using Fresnel diffraction from a phase discontinuity.

A field-portable, single-shot and very simple method is presented for measuring the full complex degree of coherence (CDC) of a quasi-monochromatic Schell-model field using the Fresnel diffraction from a phase discontinuity. To validate the proposed technique, the CDC of the light emitted from an incoherent source with variable size is investigated. The results are in excellent agreement with theoretical predictions of the Van Cittert-Zernike theorem. This technique is also applicable to characterize the coherence properties of sources in other spectral regions, e.g., x-rays.

Partially coherent vortex beams of arbitrary radial order and a van Cittert–Zernike theorem for vortices

The orbital angular momentum properties of beams having an arbitrary degree of coherence for any radial and azimuthal order are analyzed. A generalized analytic solution is determined and the vortex structure on propagation is found to reconstruct itself, an effect that can be considered as a vortex supplement to the van Cittert--Zernike theorem.

Coherent diffusion of partial spatial coherence

Partially coherent fields are abundant in many physical systems. While the propagation of partially coherent light undergoing diffraction is well understood, its evolution in the presence of coherent diffusion (i.e., diffusion of complex fields) remains largely unknown. Here we develop an analytic model describing the diffusion of partially coherent beams and study it experimentally. Our model is based on a diffusion analog of the famous Van Cittert–Zernike theorem. Experimentally, we use a four-wave mixing scheme with electromagnetically induced transparency to couple optical speckle patterns to diffusing atoms in a warm vapor. The spatial coherence properties of the speckle fields are monitored under diffusion and are compared to our model and to the familiar evolution of spatial coherence of light speckles under diffraction. We identify several important differences between the evolution dynamics of the spatial coherence under diffraction and diffusion. Our findings shed light on the propagation of partially coherent fields in media where multiple scattering or thermal motion lead to coherent diffusion.

Free evolution of a mixture of quantum states with classical space and momentum uncertainties

This article continues and completes a recent one regarding a quantum version of van Cittert–Zernike’s theorem (Fabre et al 2018 Eur. J. Phys. 39 015401). In that paper, we considered a mixture of quantum states that only differed in their initial positions, but not on their momenta. Here, we relax this nonphysical assumption and analyse how the inclusion of both classical uncertainties in the initial position and momentum of a mixture of—otherwise identical—free quantum states which do not interact with an environment, affects its quantum coherence length. We show that the mathematical development and the final result are simple enough to make possible the incorporation of this subject into a basic course of quantum physics. Furthermore, we demonstrate that these results provide a comprehensive solution to a recent controversy in atomic collision physics.

Speckle coherence of piecewise-stationary stochastic targets.

The van Cittert-Zernike (VCZ) theorem describes the propagation of spatial covariance from an incoherent source distribution, such as backscatter from stochastic targets in pulse-echo imaging. These stochastic targets are typically assumed statistically stationary and spatially incoherent with uniform scattering strength. In this work, the VCZ theorem is applied to a piecewise-stationary scattering model. Under this framework, the spatial covariance of the received echo data is demonstrated as the linear superposition of covariances from distinct spatial regions. This theory is analytically derived from fundamental physical principles, and validated through simulation studies demonstrating superposition and scaling. Simulations show that linearity is preserved over various depths and transmit apodizations, and in the presence of noise. These results provide a general framework to decompose spatial covariance into contributions from distinct regions of interest, which may be applied to advanced imaging methods. While the simulation tools used for validation are specific to ultrasound, this analysis is generally applicable to other coherent imaging applications involving stochastic targets. This covariance decomposition provides the physical basis for a recently described imaging method, Multi-covariate Imaging of Sub-resolution Targets.

Experimental observation of predictions of the generalized van Cittert–Zernike theorem for quasi-homogeneous planar electromagnetic sources

We present the experimental observation of predictions of the extended generalized van Cittert–Zernike theorem (eVCZT) for the cross-spectral density matrix of quasi-homogeneous, planar electromagnetic sources. We describe the experimental setup built to achieve this observation and discuss the results obtained in our experiments. The setup is the combination of a quasi-homogeneous, planar electromagnetic source with controllable polarization spatial distribution and a Stokes polarimeter. Our results show that the eVCZT correctly predicts the diameter, spatial coherence, and polarization properties of a beam-like field produced by a quasi-homogeneous source with controllable spectral density spatial distribution for two orthogonal linear polarization components. We found that applying the eVCZT it is possible to generate a beam-like optical field with a central spot and two shifted replicas. The three spots generated have similar values for the normalized linear polarization components, whereas the central spot and the replicas have opposite handedness for the normalized circularly polarized component. The source presented in this work is intended to be used to perform polarimetric measurements of the second-order spatial correlations in quasi-homogeneous specimens.