A comparison of diffracting and non-diffracting coherence functions based on full field spatial coherence visualization system

Entangled-photon pairs ~biphotons! generated by spontaneous optical parametric down-conversion exhibit a number of properties that are analogous to those of ordinary photons generated by incoherent sources. The spatial pump-field distribution and the two-particle wave function in the biphoton case play the respective roles of the source intensity distribution and the second-order coherence function in the incoherent case. The van Cittert‐Zernike theorem, which is applicable for incoherent optical sources emitting independent photons, has a counterpart for biphotons. Likewise, the partial-coherence theory of image formation has an analogous counterpart for biphoton beams transporting spatial information. However, an underlying duality, rather than analogy, accompanies the mathematical similarity between incoherent and biphoton emissions if the comparison is made between the photon count rate in the incoherent case and the biphoton count rate in the entangledphoton case. The smaller the size of an incoherent source, the more separable is the coherence function and the more coherent is the field, and therefore the higher the visibility of ordinary interference fringes. In contrast, the narrower the size of a biphoton pump source, the more separable is the wave function and the less entangled is the field, and therefore the lower the visibility of biphoton interference fringes. This duality is similar to the complementarity between single and two-photon interference exhibited for biphotons.

In order to estimate the statistical phase of underwater propagation for coherent acoustic systems, a calculation of background structure functions, based on wavespeeds, provides a wavefront distortion metric that excludes the phase due to focusing. In 1890, Gouy measured a phase change through the focus that differs for spherical vs cylindrical convergence by a factor of 2 from π/2. Statistics for the locations of the probable focus waist can use structure functions based on the measured sound speed profiles. Phase estimates are susceptible to errors in phase estimation unless results consider the Gouy phase while passing through focus waists. These foci may counteract or average with other errors of estimation, including scattering, type of spreading (cylindrical vs spherical foci), and reflections from the surface and bottom. Oceans have a sufficient range for the spatial derivatives of sound speed profiles to contribute persistently to foci that we show have coherence time and cross-sectional spatial (imaging) coherence. As an alternative, specular point theory can estimate deterministic pressure solutions. Convergence of these ray models that violate the stationary phase requires methods such as catastrophe theory to estimate the focal effects. Sommerfeld had an early comprehensive ray theory description that includes the Gouy phase. Using Flatté’s data, the foci are clearly time and length coherent. Statistically, the mutual coherence function for the Atlantic, Caribbean, and Gulf of Mexico regions shows that even distorted and out-of-phase convergence has full spatial coherence.

A method is presented for obtaining the far-field diffraction pattern of a plane aperture when the mutual coherence function of the illumination contains a quadratic phase term. The method is applied to obtain the diffraction pattern of a slit and circular aperture illuminated by an extended quasimonochromatic incoherent slit and circular source, respectively. The phase term has both a quantitative and qualitative effect on the features of the diffraction pattern.

Investigation of the spatial coherence function of wind loads on lattice frame structures

We have succeeded in developing a laser-pumped X-ray laser with full spatial coherence at 13.9 nm. The X-ray laser beam with a very small divergence of 0.2 mrad was generated from double target experiments, where a seeding light from the first laser medium was amplified in the second laser medium. The observed divergence was close to the diffraction limit value within a factor of two. The seeding light was amplified in the second medium without refraction influence and the gain coefficient was 7.9 cm-1. From the measurement of visibility, it was found that the spatial coherent length was longer than the beam diameter.

Models for the coherence of quasi-monochromatic light from spherical incoherent sources are constructed by placing incoherent monopole and dipole sources on the surface of a sphere. All models allow for relatively simple numerical calculations of coherence functions for arbitrary source sizes and positions of observation points. We show analytically and confirm numerically that the far-field regime for transverse coherence is formed at distances larger than the source size, regardless of the wavelength.

Acoustic wave-front curvature ranging (WCR) is a localization method that exploits the curvature of the arriving wave-front to determine the range of an acoustic emitter. By assuming that the wave-front is circular, it provides an estimate of the range as the radius of the circle. It is based on the measurement of time of arrival differences for correlated transient signals between pairs of spatially separated receivers. Performance bounds are determined by: (1) source and receiver geometry, (2) signal characteristics, and (3) spatial coherence limits due to oceanic effects. Both theoretical and Monte Carlo studies have been done which define the performance limitations as a function of aperture size, signal structure and spatial coherence. Results show that for a given aperture and fixed range accuracy, the low-frequency limit is bounded by the signal characteristics (SNR and time-bandwidth product), whereas the high-frequency limit is determined by the coherence length allowed by the ocean. This high-frequency limit is presently estimated by extrapolation of lower frequency models and is imprecisely known due to the unavailability of high-frequency, short time spatial coherence data.

The notion spatial coherence is used by sonar systems such as Correlation Velocity Logs (CVL) and Synthetic Aperture Sonars (SAS). It exploits correlation between acoustic signals sensed in space and time to estimate for example platform displacement. The Van Cittert Zernike (VCZ) theorem, is a classical statistical optics theorem that predicts the spatial coherence of a pressure field backscattered by a random medium. However it appears that intrinsic characteristics of Low Frequency Synthetic Aperture Sonar (LFSAS) can breach VCZ asumptions.In this paper, the notion of spatial coherence applied to a Low Frequency SAS system (LFSAS) is investigated. In a first part, the notion of spatial coherence and the VCZ therorem is presented. Then the High Resolution LFSAS (HRLFSAS) developed by NATO CMRE and used in this work is introduced. In the same time, a simulation tool is presented. In this paper, typical coherence functions obtained on simulation are presented. That allows to point out the link between the shape of the coherence function and the length of the transmit antenna. These functions are compared to the coherence observed on LFSAS data. Differences are investigated in terms of signal to noise ratio.

A nonuniform sampling scheme is described for measuring the mutual intensity of the wave field produced in a plane a distance z from a spatially incoherent, three-dimensional source object. Both uniform and nonuniform sampling are analyzed and discussed in detail, and comparisons of the two schemes are made. It is shown that nonuniform sampling requires fewer measurements than uniform sampling to specify the coherence function. Additionally, the smallest separation required between measurement points is larger for the nonuniform case.

Spatial coherence measurement of X-ray undulator radiation

In order to apply Kriging methods for geostatistics of spatial data, an estimation of spatial coherency functions is required priorly based on the spatial distance between measurement points. In the study, the typical coherency functions, such as semi-variogram, homeogram, and covariance function, were estimated using the national geoid model. The test area consisting of 2°×2° and the Unified Control Points (UCPs) within the area were chosen as sampling measurements of the geoid. Based on the distance between the control points, a total of 100 sampling points were grouped into distinct pairs and assigned into a bin. Empirical values, which were calculated with each of the spatial coherency functions, resulted out as a wave model of a semi-variogram for the best quality of fit. Both of homeogram and covariance functions were better fitted into the exponential model. In the future, the methods of various Kriging and the functions of estimated spatial coherency need to be studied to verify the prediction accuracy and to calculate the Mean Squared Prediction Error (MSPE).

A set of conditions is introduced for spatial coherence and temporal coherence in an interferometer to be independent, such that the coherence function appears as the product of its spatial and temporal terms. These conditions are commonly fulfilled in usual setups.

Coherence holography capable of real-time recording and reconstruction is proposed and experimentally demonstrated with a generic Leith-type coherence hologram. The coherence hologram is optically generated in real-time using a Mach-Zehnder interferometer and reconstructed using a Sagnac radial shearing interferometer. With this method one can create an optical field distribution with a desired spatial coherence function, and visualize the coherence function in real-time as the contrast and phase variations in an interference fringe pattern. The reconstructed image of the complex coherence function has been quantified with the Fourier transform method of fringe-pattern analysis.

The spatial coherence function |Γ(s)| of ruby laser pulses has been measured by self-diffraction at a laser-induced grating depending on the distances between two points in the beam profile. The spatial coherence of the laser beam was changed by operating the laser in the fundamental transverse mode, in a transverse multimode and by applying different pump powers. The degree of spatial coherence was experimentally found to be a Gaussian, as described by the Collett-Wolf model. The new method for measuring the spatial coherence function |Γ(s)| is described and possible improvements are discussed.

The behavior of a spatial coherence function is considered, depending on time, the coherence of the source radiation, and the distance to a source and its size. It is shown that as the distance grows, the source’s partial temporal coherence reduces the degree of spatial coherence and there is a change in the form of the function of spatial coherence, which also begins to depend on the choice of a point in the cross section of an X-ray beam or a pulse.