Mutual Intensity Research Articles

Propagation of transverse optical coherence in random multiple-scattering media

We experimentally investigate the evolution of the transverse spatial coherence of light as it traverses a random, multiple-scattering medium. For near-forward scattering, it has been proposed that the wave-transport process can be well described by a transport equation for the spatial-angular Wigner function of the light, or equivalently for the two-point spatial coherence function (mutual intensity). We find good agreement between the wave-transport theory and our experimental results for media of different thickness. In a dense medium (for example, optical distance, or density, greater than 5), the nature of the scattered light field can be qualitatively described by a complex Gaussian-Schell model, which raises an interesting interpretation about the process of long-path optical transport.

A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light

Abstract We combine a telescopic imaging system with a common-path, lateral-shearing interferometer and use phase-shifting interferometry to measure the complex spatial coherence function (or mutual intensity) of a linearly-polarized optical field. Our telescopic design increases the numerical aperture of the system without distorting the shape of the wave front, and therefore without changing the phase difference between lateral positions in the optical field. Our method of generating lateral-sheared images introduces no additional astigmatism. To demonstrate the use of the interferometer we extract the information contained in the complex spatial coherence function to reconstruct the amplitude and phase of a coherent optical field, and we also show how the spatial coherence function evolves from a coherent field to a partially coherent one as light traverses a random multiple-scattering medium.

A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence function of light

We combine a telescopic imaging system with a common-path, lateral-shearing interferometer and use phase-shifting interferometry to measure the complex spatial coherence function (or mutual intensity) of a linearly-polarized optical field. Our telescopic design increases the numerical aperture of the system without distorting the shape of the wave front, and therefore without changing the phase difference between lateral positions in the optical field. Our method of generating lateral-sheared images introduces no additional astigmatism. To demonstrate the use of the interferometer we extract the information contained in the complex spatial coherence function to reconstruct the amplitude and phase of a coherent optical field, and we also show how the spatial coherence function evolves from a coherent field to a partially coherent one as light traverses a random multiple-scattering medium.

Energetic efficient synthesis of general mutual intensity distribution

The mutual intensity distribution of a light beam may contain available information. The task of encoding a given mutual intensity distribution is addressed in this paper. Various approaches for encoding the mutual intensity function have been previously proposed. However, all of them provide low energetic efficiency and commonly require sophisticated production methods. The idea of using a phase-only filter for performing this synthesis is hereby investigated. The proposed method is numerically examined for the case of placing the mutual intensity generating filter in the fractional Fourier domain.

Visible cone-beam tomography with a lensless interferometric camera

Digital processing of optical coherence functions can reconstruct three-dimensional objects illuminated by incoherent light. It is shown that Fourier analysis of the mutual intensity of the field produces projections that are mathematically identical to the projections of x-ray cone-beam tomography. A lensless interferometric camera that captures planes of mutual intensity data is described and used to reconstruct an incoherently illuminated visible object in three dimensions.

Spatial coherence property of a laser beam during acousto-optic diffraction

The spatial coherence property of a partially coherent light during the Bragg acousto-optic interaction is investigated. Starting from the wave equation, four coupled, parabolic equations that can describe the evolution and the propagation of mutual intensity functions of the diffracted light during the acousto-optic interaction are derived. A partially coherent light beam with arbitrary spatial profile and complex degree of spatial coherence is assumed to be incident on the Bragg acousto-optic cell. With the use of a statistical theory of linear systems, a general formalism of angular-correlation functions for zero-order and minus-one-order light can be derived. The corresponding mutual intensity and complex coherence factor functions are hence implemented numerically. From the solutions one can note that, through the acousto-optic interaction, the degrees of spatial coherence of the diffracted light beams are controllable by the intensity and the frequency of the sound wave.

The irradiance of partially polarized beams in a scalar treatment

It is shown that the irradiance distribution produced throughout the space by a partially polarized beam can be treated in scalar terms. To this aim it is necessary to introduce an equivalent mutual intensity across a beam section. This can be made by exploiting the properties of the beam coherence-polarization matrix.

Synthesis of spatial coherence

The mutual intensity function plays a major role in characterizing quasi-monochromatic, partially coherent optical signals. We propose to use the mutual intensity as a carrier of information to avoid speckle noise in coherent illumination systems and to permit the use of complex functions that are prohibited spatially incoherent sources. To do this we require methods for encoding the information as a coherence function. An optical system for synthesizing a beam with a given mutual intensity function is proposed. The optical system permits the synthesis of any desired mutual intensity function. The illumination is supplied by a quasi-monochromatic, spatially incoherent source. Experimental results demonstrate the performance of this system for several cases.

Storage and retrieval of correlation functions of partially coherent fields

A new method is described for determining the two-point equal-time coherence function (the mutual intensity) and the two-point equal-time intensity correlation function of partially coherent fields. The method is reminiscent of conventional holography but differs from it in several important respects.

Three-dimensional coherence imaging in the Fresnel domain

We show that three-dimensional incoherent primary sources can be reconstructed from finite-aperture Fresnel-zone mutual intensity measurements by means of coordinate and Fourier transformation. The spatial bandpass and impulse response for three-dimensional imaging that result from use of this approach are derived. The transverse and longitudinal resolutions are evaluated as functions of aperture size and source distance. The longitudinal resolution of three-dimensional coherence imaging falls inversely with the square of the source distance in both the Fresnel and Fraunhofer zones. We experimentally measure the three-dimensional point-spread function by using a rotational shear interferometer.

Elastic scattering of partially coherent beams of fast electrons by a crystal with a defect.

Scattering of a quasi-monochromatic electron beam by a crystal with a defect is described with the use of the mutual coherency function and the formalism of quasi-Bloch waves. An expression correlating the mutual intensity on the exit and entrance surfaces of the crystal in terms of the scattering matrix has been found. The matrix elements are determined by a system of integro-differential equations, which have been obtained without using the column approximation. It has been shown that calculations of the matrix elements can be significantly simplified when the approximation of the small-angle scattering of quasi-Bloch waves by the defect displacement field is satisfied. Such an approximation can be applied in many cases, e.g. to a crystal with a dislocation. The mutual intensity on the crystal entrance surface has been found for the general case of defocused illumination. As an example of applying the new approach, expressions for the intensity in convergent-beam electron diffraction (CBED) and large-angle CBED (LACBED) patterns have been obtained. The LACBED patterns of a crystal with a dislocation have been simulated. It has been shown that the developed approach allows a more exact simulation of the LACBED than do the conventional approaches using the column approximation and the approximation of independent plane waves filling the illumination cone.

Display of spatial coherence

The mutual-intensity function plays a major role in characterizing quasi-monochromatic, partially coherent optical signals. We demonstrate an optical system for displaying the mutual intensity of a one-dimensional input beam. The experimental system is based on the fact that the mutual intensity of a signal can be expressed as the ensemble averaging of a cross-correlation operation between two related optical signals. The setup consists of a Sagnac interferometer followed by an optoelectronic joint transform correlator. Experimental results demonstrate the capabilities of the mutual-intensity analyzer.

Performance bounds for high-light-level amplitude and intensity interferometry

Amplitude interferometry and intensity interferometry are methods for processing the complex amplitude and the intensity, respectively, of an electromagnetic field to estimate the mutual intensity of the field at two spatial locations. Whereas complex amplitude measurements allow for the direct estimation of the mutual-intensity phase, intensity measurements do not. We consider applications for which the estimation of the magnitude or the squared magnitude of the mutual intensity is adequate, and we provide fundamental limits on the estimation accuracy of any unbiased estimator of the squared magnitude of the mutual intensity from coherent (complex amplitude) or incoherent (intensity) measurements. Our analysis is performed for the high-light-level (classical-noise-limit) case and quantifies the advantages of making the more difficult coherent measurements.

Matrix treatment for partially polarized, partially coherent beams

A matrix method is outlined for dealing with quasi-monochromatic, partially polarized light when spatial coherence is not necessarily complete and propagation occurs along beams. Both spatial coherence and polarization properties are described by a single 2x2 matrix whose elements have the structure of mutual intensity functions. Through a simple example it is shown that this matrix can account for differences that would not be revealed by a scalar treatment or by a locally defined polarization matrix.

Analytic relation for recovering the mutual intensity by means of intensity information

An analytic relation for recovering the mutual intensity by means of intensity information under the condition of the fractional Fourier transform is derived. The results may simplify the reconstruction of the mutual intensity in comparison with the Wigner tomographic method and can be regarded as an inverse transform formula that expresses output intensity in terms of input mutual intensity under partially coherent illumination.

Is the Van Cittert-Zernike Theorem Applicable to Spatially Incoherent Broadband Spectral Source?

This paper deals with the propagation law of mutual intensity and cross-spectral density of the light emitted from an extended broadband spectral source that is spatially incoherent. The mutual intensity of light does not obey the same mathematical formalism as the Van Cittert-Zernike theorem applicable to quasi-monochromatic light, although the cross-spectral density of light does in the propagation zone of the Fresnel diffraction region.

Scattering from a fluctuating object in a fluctuating waveguide

In many practical applications of active target scattering in ocean acoustics, there is a significant amount of uncertainty in both the scattering properties of the target and the propagation characteristics of the medium. In order to consistently combine a reasonable description of this uncertainty with the established physics of scattering from submerged objects, a statistical model is developed that describes both the spatial and temporal characteristics of the scattered field. An essential aspect of this model is the multiplicative nature of the scattering process. The acoustic field propagates from the source through a random waveguide. It is then scattered from a random target that undergoes fluctuations that are uncorrelated with those of the incident field. This scattered field then propagates into azimuthal directions where the random waveguide to the receiver may be partially or fully uncorrelated with the incident waveguide. Standard assumptions for statistical fluctuations of the waveguide channels and target lead to a non-Gaussian field at the receiver. Possible randomness of the sound source leads to a further departure from Gaussianity in the received field statistics. Expressions for the spatio-temporal mutual intensity structure of the received field will be provided, with illustrative examples for a shallow water scenario.

Spatial coherence properties of a multiple aperture system an analysis based on the Walsh functions

Analysis of the spatial coherence of the light transmitted by an optical device composed of a periodical array of identical apertures is developed by employing an approach based on the properties of the binary Walsh functions. The successive interactions between each aperture, and the mutual intensity characterizing the coherence state of the transmitted light, can be adequately explained through the behaviour of the Walsh-Hadamard spectrum associated with the intensity distribution resulting from the far-field propagated light at the output of the aperture array.

A study on spatial coherence using quadratic radially distributed apertures (application to confocal imaging)

We study the spatial coherence problem using an amplitude modulation applied to confocal imaging systems. This type of modulation assumes a quadratic radial distribution. The mutual coherence intensity or the coherence factor is calculated and compared with the results obtained for clear circular apertures.

Wave field determination using tomography of the ambiguity function

Ambiguity function (AF) theory is proposed to reconstruct a complex wave field using tomography by measurements of intensity in a refractive optical system. By performing one-dimensional (1D) inverse Fourier transforms of intensities with some adjustment in various longitudinal optical system parameters, the corresponding AF values along the lines at different angles in the AF phase space are obtained; therefore, the mutual intensity function is reconstructed by performing a 1D Fourier transform of the reconstructed AF values. The reconstruction process in some cases is considered to be simpler than the equivalent theory using Wigner distribution function.