By Adam J. Schwartz, Mukul Kumar, Brent L. Adams, David P. Field

Crystallographic texture or hottest orientation has lengthy been identified to strongly impression fabric homes. traditionally, the technique of acquiring such texture info has been even though using x-ray or neutron diffraction for bulk texture measurements, or transmission electron microscopy (TEM) or electron channeling for neighborhood crystallographic details. in recent times, we've seen the emergence of a brand new characterization strategy for probing the microtexture of fabrics. This boost has take place essentially during the computerized indexing of electron backscatter diffraction (EBSD) styles. the 1st commercially to be had approach was once brought in 1994, and because then the expansion of revenues world wide has been dramatic. This has observed widening applicability in fabrics technology difficulties resembling microtexture, section identity, grain boundary personality distribution, deformation microstructures, and so forth. and is proof that this method can, on occasion, exchange extra time-consuming TEM or X-ray diffraction investigations. the aim of this e-book is to supply the elemental foundation for EBSD. The formation and interpretation of EBSD styles and the gnomonic projection are defined because the framework for fabrics characterization utilizing EBSD. conventional illustration of texture in orientation house is mentioned by way of stereographic projections, pole figures, inverse pole figures, and orientation distribution capabilities sooner than introducing the Rodrigues-Frank illustration of crystallographic texture. the basics of computerized EBSD and the accuracy of EBSD measurements are then mentioned. present and software program in addition to destiny customers for studying EBSD information units are reviewed. a short point out of the criterion required for the acquisition of an EBSD process is integrated as an reduction to this fairly new sector of fabrics characterization. The part concludes with chapters from 3 brands of EBSD gear that spotlight fresh advances in features. The publication concludes with a evaluation of contemporary purposes of the strategy to clear up tough difficulties in fabrics technological know-how in addition to demonstrates the usefulness of coupling EBSD with different ways comparable to numerical research, plasticity modeling, and TEM. cognizance is paid to the size and mapping of pressure utilizing EBSD in addition to the characterization of deformed microstructures, non-stop recrystallization, research of aspects, ceramics, and superconducting fabrics.

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177) that has been multiplied by xl0 , then Hu is eliminated and one obtains Er ¼   Àjb @ xl0 @ E H þ z z : b r@u x2 l0 e À b2 @r ð2:183Þ Analogous eliminations lead to Eu ¼  Àjb x2 l0 e À b 2  @ xl @ Ez À 0 Hz ; r@u b @r   Àjb xe @ @ Ez þ Hz ; À Hr ¼ b r@u @r x 2 l 0 e À b2   Àjb xe @ @ Ez þ Hz : Hu ¼ r@u x2 l0 e À b2 b @r ð2:184Þ ð2:185Þ ð2:186Þ So, it is sufficient to solve for Ez and Hz because the other field components may be derived from these. We do this with the wave equation, separately for core and cladding regions due to the different refractive indexes.

The eigenvalue Eq. 206) is quadratic in Jl0 ðuÞ ðuJl ðuÞÞ. 203) results in larger values of jA=Bj. These are called HElm modes. For the positive sign, EHlm modes result, with smaller values of jA=Bj. HElm and EHlm modes are hybrid modes, where the boundary conditions can be fulfilled only by the presence of both longitudinal field components. All six field components exist. For comparison: Cylindrical hollow waveguides have Hlm and Elm modes where one of Ez , Hz vanishes. The radial dependence of the field amplitudes of the HE11 mode is well approximated by a fundamental Gaussian beam.

The two right hand sides must also be equal, 1 which results in B=C ¼ Æ1. 165) ) AeÀv ¼ 2B cos u ¼ DeÀv , A=D ¼ 1. 164) ) À n1 n2 vAeÀv ¼ À2uB sin u. We divide this by AeÀv ¼ 2B cos u and obtain À  Á À  Á v ¼ n22 n21 u tan u, B=A ¼ eÀv =ð2 cos uÞ ¼ n21 n22 ðveÀv Þ=ð2u sin uÞ. À  Á Case B ¼ ÀC: AeÀv ¼ 2jB sin u ¼ ÀDeÀv , A=D ¼ À1, À n21 n22 vAeÀv À  Á À  Á ¼ 2juB cos u, v ¼ À n22 n21 u cot u, B=A ¼ eÀv =ð2j sin uÞ ¼ n21 n22 ðjveÀv Þ =ð2u cos uÞ. 2 Dielectric Waveguides pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À2 u tanðu À mp=2Þ ¼ nÀ2 V 2 À u2 12 v ¼ n12 33 eigenvalue equation for E modes: ð2:168Þ where n12 ¼ n2 =n1 is the relative refractive index change from medium 1 to medium 2.

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