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A theory for the characterization of the fourth-order moment of electromagnetic wave beams is presented in the case when the source is partially coherent. A Gaussian–Schell model is used for the partially coherent random source. The white-noise paraxial regime is considered, which holds when the wavelength is much smaller than the correlation radius of the source, the beam radius of the source, and the correlation length of the medium, which are themselves much smaller than the propagation distance. The complex wave amplitude field can then be described by the Itô-Schrödinger equation. This equation gives closed evolution equations for the wave field moments at all orders and here the fourth-order moment equations are considered. The general fourth-order moment equations are solved explicitly in the scintillation regime (when the correlation radius of the source is of the same order as the correlation radius of the medium, but the beam radius is much larger) and the result gives a characterization of the intensity covariance function. The form of the intensity covariance function derives from the solution of the transport equation for the Wigner distribution associated with the second-order wave moment. The fourth-order moment results for polarized waves are used in an application for imaging of partially coherent sources.
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SegmentGeometry: A Tool for Measuring Second Moment of Area in 3D Slicer
Synopsis Second moment of area is a measure of how well the cross-section of a beam will resist bending because of its shape. Many have used second moment of area to investigate the mechanical adaptations of biological structures from stingray jaws to animal limb bones. In this context it is important to acknowledge the assumptions of beam theory, in which second moment of area plays a key role, if reasonable results are desired. For example, to minimize shear the structure should be at least 10 times longer than it is wide and deflection should be minimal. Analyzing the internal geometry of biological structures has never been easier or more accessible given the wide, and growing availability of micro-CT scans. Here, we offer a guide on the care that needs to be taken when interpreting second moment of area, and present open-access, open-source software that can process hundreds if not thousands of structures in a short time frame. SegmentGeometry, an extension for the open-source imaging platform 3D Slicer, iterates slice-by-slice through 3D structures to calculate second moment of area and other cross-sectional properties. We analyzed 2 case studies to demonstrate the power of this tool and to highlight interpretations that can be gleaned from second moment of area. Second moment of area is just one part of the Euler–Bernoulli beam theory and considering the full equation would greatly increase the number and diversity of questions that can be answered.
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- Award ID(s):
- 1759637
- PAR ID:
- 10363902
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Integrative Organismal Biology
- Volume:
- 4
- Issue:
- 1
- ISSN:
- 2517-4843
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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