Optimization Of Waveguide Optics For Lensless X Ray Imaging

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Optimization of Waveguide Optics for Lensless X-ray Imaging

Author : Sven Philip Krüger
Publisher : Universitätsverlag Göttingen
ISBN 13 : 3863950151
Total Pages : 169 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis Optimization of Waveguide Optics for Lensless X-ray Imaging by : Sven Philip Krüger

Download or read book Optimization of Waveguide Optics for Lensless X-ray Imaging written by Sven Philip Krüger and published by Universitätsverlag Göttingen. This book was released on 2011 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: Lensless x-ray imaging is a promising method to determine the three-dimensional structure of material science and biological specimens at the nanoscale. The development of this technique is strongly related to the optimization of x-ray optics since the image formation and object reconstruction depend significantly on the properties of the illumination wave-field. Waveguide optics act as quasi-point sources and enable the spatial and coherent filtering of x-ray beams. Up to now, x-ray waveguides were severely limited in transmission and flux, restricting their use to high-contrast test structures with moderate resolution and long accumulation times. To overcome these limitations, a novel waveguide design with an optimized refractive index profile is presented which significantly minimizes the absorption of the modes propagating inside the waveguide. Experimental results along with simulations show that these two-component planar x-ray waveguides provide small beam cross-sections along with a high photon flux at the exit. By a serial arrangement of two waveguide slices an optimized illumination source has been developed for high-resolution microscopy, as demonstrated in proof-of-concept imaging experiments.

X-ray waveguide optics

Author : Sarah Hoffmann-Urlaub
Publisher : Göttingen University Press
ISBN 13 : 3863953088
Total Pages : 134 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis X-ray waveguide optics by : Sarah Hoffmann-Urlaub

Download or read book X-ray waveguide optics written by Sarah Hoffmann-Urlaub and published by Göttingen University Press. This book was released on 2017 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: Modern x-ray sources and analysis techniques such as lens less imaging combined with phase retrieval algorithms allow for resolving structure sizes in the nanometer range. For this purpose optics have to be employed, ensuring small focal spot dimensions simultaneously with high photon densities. Furthermore, the wave front behind the optics is required to be smooth enabling for high resolution imaging. Combining all these properties, x-ray waveguides are well suited to perform this task, since the intensity distribution behind the guide is restricted in two dimensions serving as a secondary quasi point-source without wave-front aberrations, showing also a high divergence, suitable for resolving fine features. Importantly, the radiation provided by the waveguide reveals a high degree of coherence, required by many imaging techniques. The waveguide itself consists of an air-filled channel embedded in a solid matrix; typical materials are silicon, germanium or quartz. While the entrance area is nano-sized, the channel length is in the millimeter-range, this way posing challenges to fabricate high aspect ratio geometries. Since the functioning of x-ray waveguides is based on the total reflection at small incident angles, the surface roughness of the channel walls must be as low as possible to avoid scattering and hence loss of intensity. To fulfill these demanding conditions, a process scheme involving spin-coating, electron beam lithography, wet development, reactive ion etching and wafer bonding is optimized within this work. To gain deeper insights into the principle of wave guiding finite difference simulations are performed, also opening access for advanced design considerations such as gratings, tapered and curved channels, or beamsplitters, enabling for constructing novel x-ray tools as for example time delay devices or interferometers. Waveguides in all geometries are tested at synchrotron sources, accomplishing new benchmarks in x-ray optical performance. Here, the x-ray beam leaving the channel, propagates out to a pixel array detector in the far-field region. From the recorded data the intensity distribution in the near-field directly behind the waveguide is reconstructed, revealing an outstanding agreement with the simulations and electron micrographs. Since the radiation field of the waveguide is well-characterized and also tunable to meet the requirements of both the measurement setup and the sample, they are suited of a broad field of applications in coherent x-ray imaging.

A Dedicated Endstation for Waveguide-based X-ray Imaging

Author : Sebastian Kalbfleisch
Publisher : Universitätsverlag Göttingen
ISBN 13 : 3863951018
Total Pages : 188 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis A Dedicated Endstation for Waveguide-based X-ray Imaging by : Sebastian Kalbfleisch

Download or read book A Dedicated Endstation for Waveguide-based X-ray Imaging written by Sebastian Kalbfleisch and published by Universitätsverlag Göttingen. This book was released on 2013 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Advanced x-ray multilayer waveguide optics

Author : Qi Zhong
Publisher : Göttingen University Press
ISBN 13 : 3863953258
Total Pages : 164 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis Advanced x-ray multilayer waveguide optics by : Qi Zhong

Download or read book Advanced x-ray multilayer waveguide optics written by Qi Zhong and published by Göttingen University Press. This book was released on 2017 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis was to design novel waveguide structures, and to analyze them in view of complex phenomena of near-field propagation. For this purpose, experimental far-field measurements were used in combination with finite-difference simulations and phase retrieval methods. Two novel structures have been designed, fabricated and characterized: the waveguide array (WGA), yielding several waveguided beams in transmission, and multi-guide resonate beam couplers (RBCs), tailored to yield two or several reflected beams. Two novel structures have been designed, fabricated and characterized: the WGA, yielding several waveguided beams in transmission, and multi-guide RBCs, tailored to yield two or several reflected beams. The WGA and the multi-guide RBCs are not only distinct in the coupling geometry. A major difference is related to the fact that the WGA principle is based on the separation (non coupling) of the different transmitted wavelets, while the RBC functions are based on a strong coupling of guided radiation in several layers.

Cone-beam X-ray Phase Contrast Tomography of Biological Samples

Author : Matthias Bartels
Publisher : Universitätsverlag Göttingen
ISBN 13 : 3863951344
Total Pages : 220 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis Cone-beam X-ray Phase Contrast Tomography of Biological Samples by : Matthias Bartels

Download or read book Cone-beam X-ray Phase Contrast Tomography of Biological Samples written by Matthias Bartels and published by Universitätsverlag Göttingen. This book was released on 2013 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: Three-dimensional information of entire objects can be obtained by the remarkable technique of computed tomography (CT). In combination with phase sensitive X-ray imaging high contrast for soft tissue structures can be achieved as opposed to CT based on classical radiography. In this work biological samples ranging from micrometer sized single cells over multi-cellular nerve tissue to entire millimeter sized organs are investigated by use of cone-beam propagationbased X-ray phase contrast. Optimization with respect to contrast, resolution and field of view is achieved by addressing instrumentation, sample preparation and phase reconstruction techniques. By using laboratory sources functional soft tissue within the bony capsule of mouse cochleae is visualized in 3D with unprecedented image quality. At synchrotron storage rings the technique reveals more than 1000 axons running in parallel within a mouse nerve and enables doseefficient three-dimensional cellular imaging as well as two-dimensional imaging at high resolutions below 50 nm.

Biomedical Imaging

Author : Tim Salditt
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110426692
Total Pages : 358 pages
Book Rating : 4.1/5 (14 download)

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Book Synopsis Biomedical Imaging by : Tim Salditt

Download or read book Biomedical Imaging written by Tim Salditt and published by Walter de Gruyter GmbH & Co KG. This book was released on 2017-10-23 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering both physical as well as mathematical and algorithmic foundations, this graduate textbook provides the reader with an introduction into modern biomedical imaging and image processing and reconstruction. These techniques are not only based on advanced instrumentation for image acquisition, but equally on new developments in image processing and reconstruction to extract relevant information from recorded data. To this end, the present book offers a quantitative treatise of radiography, computed tomography, and medical physics. Contents Introduction Digital image processing Essentials of medical x-ray physics Tomography Radiobiology, radiotherapy, and radiation protection Phase contrast radiography Object reconstruction under nonideal conditions

Coherent X-ray diffractive imaging on the single-cell-level of microbial samples

$Download Coherent X-ray diffractive imaging on the single-cell-level of microbial samples PDF Online Free$

Author : Robin Niklas Wilke
Publisher : Göttingen University Press
ISBN 13 : 3863951905
Total Pages : 254 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis Coherent X-ray diffractive imaging on the single-cell-level of microbial samples by : Robin Niklas Wilke

Download or read book Coherent X-ray diffractive imaging on the single-cell-level of microbial samples written by Robin Niklas Wilke and published by Göttingen University Press. This book was released on 2015 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since its first experimental demonstration in 1999, Coherent X-Ray Diffractive Imaging has become one of the most promising high resolution X-Ray imaging techniques using coherent radiation produced by brilliant synchrotron storage rings. The ability to directly invert diffraction data with the help of advanced algorithms has paved the way for microscopic investigations and wave-field analyses on the spatial scale of nanometres without the need for inefficient imaging lenses. X-Ray phase contrast which is a measure of the electron density is an important contrast mode of soft biological specimens. For the case of many dominant elements of soft biological matter, the electron density can be converted into an effective mass density offering a unique quantitative information channel which may shed light on important questions such as DNA compaction in the bacterial nucleoid through ‚weighing with light‘. In this work X-Ray phase contrast maps have been obtained from different biological samples by exploring different methods. In particular, the techniques Ptychography and Waveguide-Holographic-Imaging have been used to obtain twodimensional and three-dimensional mass density maps on the single-cell-level of freeze-dried cells of the bacteria Deinococcus radiodurans, Bacillus subtilis and Bacillus thuringiensis allowing, for instance, to estimate the dry weight of the bacterial genome in a near native state. On top of this, reciprocal space information from coherent small angle X-Ray scattering (cellular Nano-Diffraction) of the fine structure of the bacterial cells has been recorded in a synergistic manner and has been analysed down to a resolution of about 2.3/nm exceeding current limits of direct imaging approaches. Furthermore, the dynamic range of present detector technology being one of the major limiting factors of ptychographic phasing of farfield diffraction data has been significantly increased. Overcoming this problem for the case of the very intense X-Ray beam produced by Kirkpatrick-Baez mirrors has been explored by using semi-transparent central stops.

3d virtual histology of neuronal tissue by propagation-based x-ray phase-contrast tomography

Author : Mareike Töpperwien
Publisher : Göttingen University Press
ISBN 13 : 3863953649
Total Pages : 286 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis 3d virtual histology of neuronal tissue by propagation-based x-ray phase-contrast tomography by : Mareike Töpperwien

Download or read book 3d virtual histology of neuronal tissue by propagation-based x-ray phase-contrast tomography written by Mareike Töpperwien and published by Göttingen University Press. This book was released on 2018 with total page 286 pages. Available in PDF, EPUB and Kindle. Book excerpt: Deciphering the three-dimensional (3d) cytoarchitecture of neuronal tissue is an important step towards understanding the connection between tissue function and structure and determining relevant changes in neurodegenerative diseases. The gold standard in pathology is histology, in which the tissue is examined under a light microscope after serial sectioning and subsequent staining. It is an invasive and labor-intensive technique which is prone to artifacts due to the slicing procedure. While it provides excellent results on the 2d slices, the 3d anatomy can only be determined after aligning the individual sections, leading to a non-isotropic resolution within the tissue. X-ray computed tomography (CT) offers a promising alternative due to its potential resolution and large penetration depth which allows for non-invasive imaging of the sample's 3d density distribution. In classical CT, contrast formation is based on absorption of the x-rays as they pass through the sample. However, weakly absorbing samples like soft tissue from the central nervous system give nearly no contrast. By exploiting the much stronger phase shifts for contrast formation, which the sample induces in a (partially) coherent wavefront, it can be substantially increased. During free-space propagation behind the sample, these phase shifts are converted to a measurable intensity image by interference of the disturbed wave fronts. In this thesis, 3d virtual histology is performed by means of propagation-based x-ray phase-contrast tomography on tissue from the central nervous system of humans and mice. A combination of synchrotron-based and laboratory setups is used to visualize the 3d density distribution on varying lengths scales from the whole organ down to single cells. By comparing and optimizing different preparation techniques and phase-retrieval approaches, even sub-cellular resolution can be reached in mm-sized tissue blocks. The development of an automatic cell segmentation workflow provides access to the 3d cellular distribution within the tissue, enabling the quantification of the cellular arrangement and allowing for extensive statistical analysis based on several thousands to millions of cells. This paves the way for biomedical studies aimed at changes in cellular distribution, e.g., in the course of neurodegenerative diseases such as multiple sclerosis, Alzheimer's disease or ischemic stroke.

Cone-beam x-ray phase-contrast tomography for the observation of single cells in whole organs

Author : Martin Krenkel
Publisher : Göttingen University Press
ISBN 13 : 3863952510
Total Pages : 238 pages
Book Rating : 4.8/5 (639 download)

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Book Synopsis Cone-beam x-ray phase-contrast tomography for the observation of single cells in whole organs by : Martin Krenkel

Download or read book Cone-beam x-ray phase-contrast tomography for the observation of single cells in whole organs written by Martin Krenkel and published by Göttingen University Press. This book was released on 2015 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: X-ray imaging enables the nondestructive investigation of interior structures in otherwise opaque samples. In particular the use of computed tomography (CT) allows for arbitrary virtual slices through the object and 3D information about intricate structures can be obtained. However, when it comes to image very small structures like single cells, the classical CT approach is limited by the weak absorption of soft-tissue. The use of phase information, encoded in measureable intensity images by free-space propagation of coherent x-rays, allows a huge increase in contrast, which enables 3D reconstructions at higher resolutions. In this work the application of propagation-based phase-contrast tomography to lung tissue samples is demonstrated in close to in vivo conditions. Reconstructions of the lung structure of whole mice at down to 5 μm resolution are obtained at a selfbuilt CT setup, which is based on a liquid-metal jet x-ray source. To reach even higher resolutions, synchrotron radiation in combination with suitable holographic phase-retrieval algorithms is employed. Due to optimized cone-beam geometry, field of view and resolution can be varied over a wide range of parameters, so that information on different length scales can be achieved, covering several millimeters field of view down to a 3D resolution of 50 nm. Thus, the sub-cellular 3D imaging of single cells embedded in large pieces of tissue is enabled, which paves the way for future biomedical research.

X-ray Waveguides and Waveguide-based Lensless Imaging

Author : Christian Fuhse
Publisher :
ISBN 13 :
Total Pages : 146 pages
Book Rating : 4.:/5 (255 download)

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Book Synopsis X-ray Waveguides and Waveguide-based Lensless Imaging by : Christian Fuhse

Download or read book X-ray Waveguides and Waveguide-based Lensless Imaging written by Christian Fuhse and published by . This book was released on 2006 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Nanoscale Photonic Imaging

Author : Tim Salditt
Publisher : Springer Nature
ISBN 13 : 3030344134
Total Pages : 634 pages
Book Rating : 4.0/5 (33 download)

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Book Synopsis Nanoscale Photonic Imaging by : Tim Salditt

Download or read book Nanoscale Photonic Imaging written by Tim Salditt and published by Springer Nature. This book was released on 2020-06-09 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book, edited and authored by a team of world-leading researchers, provides a broad overview of advanced photonic methods for nanoscale visualization, as well as describing a range of fascinating in-depth studies. Introductory chapters cover the most relevant physics and basic methods that young researchers need to master in order to work effectively in the field of nanoscale photonic imaging, from physical first principles, to instrumentation, to mathematical foundations of imaging and data analysis. Subsequent chapters demonstrate how these cutting edge methods are applied to a variety of systems, including complex fluids and biomolecular systems, for visualizing their structure and dynamics, in space and on timescales extending over many orders of magnitude down to the femtosecond range. Progress in nanoscale photonic imaging in Göttingen has been the sum total of more than a decade of work by a wide range of scientists and mathematicians across disciplines, working together in a vibrant collaboration of a kind rarely matched. This volume presents the highlights of their research achievements and serves as a record of the unique and remarkable constellation of contributors, as well as looking ahead at the future prospects in this field. It will serve not only as a useful reference for experienced researchers but also as a valuable point of entry for newcomers.

Optimization of Optical Waveguides--Electro-Optic Studies

Author : R. D. Maurer
Publisher :
ISBN 13 :
Total Pages : 30 pages
Book Rating : 4.:/5 (227 download)

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Book Synopsis Optimization of Optical Waveguides--Electro-Optic Studies by : R. D. Maurer

Download or read book Optimization of Optical Waveguides--Electro-Optic Studies written by R. D. Maurer and published by . This book was released on 1973 with total page 30 pages. Available in PDF, EPUB and Kindle. Book excerpt: Experimental measurements of the spatial and temporal transfer of power of a 225 meter length of low-loss optical waveguide have been made. In particular, measurement of the angular attenuation showed substantial loss of the high order modes which reflected itself in an approximately 8.2 nsec/km decrease in measured dispersion. Additionally there was a reduction of the effective numerical aperture from 0.15 to 0.12. Negligible mode coupling was observed in this particular waveguide which allowed a phenomenological calculation of temporal output for an assumed uniform excitation of all modes. This agreed well with experimental measurements. Calculation of this output from knowledge of the index profile is presently not in agreement and some possible reasons are indicated. The first measurements of the temperature dependence of attenuation were made to determine the approximate size of the effect. Only small changes were detected up to about 400C. (Author).

Optical Waveguide Theory

Author : A.W. Snyder
Publisher : Springer Science & Business Media
ISBN 13 : 1461328136
Total Pages : 742 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis Optical Waveguide Theory by : A.W. Snyder

Download or read book Optical Waveguide Theory written by A.W. Snyder and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 742 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is intended to provide an in-depth, self-contained, treatment of optical waveguide theory. We have attempted to emphasize the underlying physical processes, stressing conceptual aspects, and have developed the mathematical analysis to parallel the physical intuition. We also provide comprehensive supplementary sections both to augment any deficiencies in mathematical background and to provide a self-consistent and rigorous mathematical approach. To assist in. understanding, each chapter con centrates principally on a single idea and is therefore comparatively short. Furthermore, over 150 problems with complete solutions are given to demonstrate applications of the theory. Accordingly, through simplicity of approach and numerous examples, this book is accessible to undergraduates. Many fundamental topics are presented here for the first time, but, more importantly, the material is brought together to give a unified treatment of basic ideas using the simplest approach possible. To achieve such a goal required a maturation of the subject, and thus the text was intentionally developed over a protracted period of the last 10 years.

Optical Waveguide Theory by the Finite Element Method

Author : Masanori Koshiba
Publisher : Springer
ISBN 13 : 9789401047135
Total Pages : 0 pages
Book Rating : 4.0/5 (471 download)

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Book Synopsis Optical Waveguide Theory by the Finite Element Method by : Masanori Koshiba

Download or read book Optical Waveguide Theory by the Finite Element Method written by Masanori Koshiba and published by Springer. This book was released on 2012-11-07 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent advances in the field of guided-wave optics, such as fiber optics and integrated optics, have included the introduction of arbitrarily-shaped optical waveguides which, in many cases, also happened to be arbitrarily inhomogeneous, dissipative, anisotropic, and/or nonlinear. Most of such cases of waveguide arbitrariness do not lend themselves to analytical so lutions; hence, computational tools for modeling and simulation are es sential for successful design, optimization, and realization of the optical waveguides. For this purpose, various numerical techniques have been de veloped. In particular, the finite element method (FEM) is a powerful and efficient tool for the most general (i. e. , arbitrarily-shaped, inhomogeneous, dissipative, anisotropic, and nonlinear) optical waveguide problem. Its use in industry and research is extensive, and indeed it could be said that with out it many optical waveguide problems would be incapable of solution. This book is intended for students, engineers, designers, and techni cal managers interested in a detailed description of the FEM for optical waveguide analysis. Starting from a brief review of electromagnetic theory, the first chapter provides the concepts of the FEM and its fundamentals. In addition to conventional elements, i. e. , line elements, triangular elements, tetrahedral elements, ring elements, and triangular ring elements which are utilized for one-dimensional, two-dimensional, three-dimensional, axisymmetric two dimensional, and axisymmetric three-dimensional problems, respectively, special-purpose elements, such as isoparametric elements, edge elements, infinite elements, and boundary elements, are also introduced.

Advanced X-ray Multilayer Waveguide Optics

Author : Qi Zhong
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (1 download)

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Book Synopsis Advanced X-ray Multilayer Waveguide Optics by : Qi Zhong

Download or read book Advanced X-ray Multilayer Waveguide Optics written by Qi Zhong and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis is to demonstrate that x-ray waveguide optics can be gen- eralized from a single guiding film to an array of planar waveguides, enabling more complex and controllable phenomena of field propagation both in partic- ular in the near-field in the vicinity of objects to be probed by coherent imaging. Two advanced x-ray multilayer waveguides (MWGs) structures, i.e. the waveg- uide array (WGA) and the multi-guide resonant beam couplers (RBCs) are de- signed and discussed. Starting from basic theoretical analysis, the structural model of MWGs is built up. Then the MWGs are s...

Optical Waveguide Sensing and Imaging

Author : Wojtek J. Bock
Publisher : Springer
ISBN 13 : 9789048116522
Total Pages : 269 pages
Book Rating : 4.1/5 (165 download)

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Book Synopsis Optical Waveguide Sensing and Imaging by : Wojtek J. Bock

Download or read book Optical Waveguide Sensing and Imaging written by Wojtek J. Bock and published by Springer. This book was released on 2009-09-03 with total page 269 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book explores various aspects of existing and emerging fiber and waveguide optics sensing and imaging technologies including recent advances in nanobiophotonics. The focus is both on fundamental and applied research as well as on applications in civil engineering, biomedical sciences, environment, security and defence. The book aims to provide a reference of state-of-the-art overviews covering a variety of topics on the interface of engineering and biomedical sciences.

Optical Waveguide Theory by the Finite Element Method

Author : Masanori Koshiba
Publisher : Springer
ISBN 13 : 9789401116343
Total Pages : 0 pages
Book Rating : 4.1/5 (163 download)

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Book Synopsis Optical Waveguide Theory by the Finite Element Method by : Masanori Koshiba

Download or read book Optical Waveguide Theory by the Finite Element Method written by Masanori Koshiba and published by Springer. This book was released on 2014-01-14 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent advances in the field of guided-wave optics, such as fiber optics and integrated optics, have included the introduction of arbitrarily-shaped optical waveguides which, in many cases, also happened to be arbitrarily inhomogeneous, dissipative, anisotropic, and/or nonlinear. Most of such cases of waveguide arbitrariness do not lend themselves to analytical so lutions; hence, computational tools for modeling and simulation are es sential for successful design, optimization, and realization of the optical waveguides. For this purpose, various numerical techniques have been de veloped. In particular, the finite element method (FEM) is a powerful and efficient tool for the most general (i. e. , arbitrarily-shaped, inhomogeneous, dissipative, anisotropic, and nonlinear) optical waveguide problem. Its use in industry and research is extensive, and indeed it could be said that with out it many optical waveguide problems would be incapable of solution. This book is intended for students, engineers, designers, and techni cal managers interested in a detailed description of the FEM for optical waveguide analysis. Starting from a brief review of electromagnetic theory, the first chapter provides the concepts of the FEM and its fundamentals. In addition to conventional elements, i. e. , line elements, triangular elements, tetrahedral elements, ring elements, and triangular ring elements which are utilized for one-dimensional, two-dimensional, three-dimensional, axisymmetric two dimensional, and axisymmetric three-dimensional problems, respectively, special-purpose elements, such as isoparametric elements, edge elements, infinite elements, and boundary elements, are also introduced.