ntial equations that govern light propagation in Kerr media are derived when the apectric field direction and magnitude vary along the light path. Case studies predict Kerr optic fringe patferns for the specific case of pointlplane electrodes. We apply the characirections theory of photoelasticity to understand these fringes. We also study birefrinia with small Kerr constant, in particular transformer oil. For this case we show that ions in the characteristic parameter theory is possible, resulting in simple integral res between the characteristic parameters and the applied electric field. We use these nships to extend the ac modulation method to measure the characteristic paramall Kerr constant media. Measurements of the characteristic parameters using the I 1070-9878/98/ $3.00 0 1998 IEEE Authorized licensed use limited to: IEEE Xplore. Downloaded on April 1, 2009 at 13:19 from IEEE Xplore. Restrictions apply.Here d,(z) and d y ( z ) are the components of d(z), e,(z) and ey(z) are the components of e(.) and ezj (2) are the components of the permittivity tensor. For lossless Kerr media ~i j ( . ) are real and symmetric. Beside the fixed zyz frame it is convenient to work in the frame that rotates with the applied electric field in which the Kerr effect is expressed. This frame will be referred as the ET frame (see Figure 2)
Kerr electro-optic fringe patterns have long been used to study space charge injection and transport phenomena in highly birefringent materials such as nitrobenzene. Most past experimental work has been limited to 1 or 2-dimensional geometries where the electric field magnitude and direction have been constant along the light path such as two concentric or parallel cylinders or parallel plate electrodes. For these geometries the extrema in the fringe patterns can be used directly to find the electric field magnitude and direction. In this work we extend the fringe based Kerr electro-optic measurement technique to a pointlplane electrode geometry which often is used in HV research to create large electric fields for charge injection at known location and at reasonable voltages. We calculate theoretical Kerr electro-optic fringe patterns for this pointlplane electrode geometry, with and without space charge distributions, for which the electric field magnitude and direction vary along the light path. We particularly compare the calculated space charge free optical patterns for the pointlplane electrodes to the optical patterns of the 2-dimensional analog bladelplane geometry. We underline the differences and study how these fringe patterns can be used to reconstruct the axisymmetric electric field components in practice.
Recently we used the onion peeling method to reconstruct the axisymmetric electric field distribution of point/plane electrodes from Kerr electro-optic measurements. The method accurately reconstructed the electric field from numerically generated data. However in the presence of experimental noise the performance was less satisfactory. The measurements were especially noisy and unstable near the needle tip which is also the interesting region since most charge injection initiates here. We develop a new algorithm for Kerr electro-optic reconstruction of space charge in axisymmetric poidplane electrode geometries. The algorithm is built on the finite element method (FEM) for Poisson's equation and will be called finite element based Kerr electro-optic reconstruction (FEBKER) hereafter. FEBKER calculates the space charge density directly to avoid the numerical problems associated with taking the divergence of the electric field, uses single parameter light intensity measurements to enable transient analysis, which otherwise is difficult since multiple parameter intensity measurements are slow due to the rotation of polarizers, and is capable of reconstruction even when the number and/or position of measurements are limited by the electrodes andlor the experimental setup. The performance of the algorithm is tested on synthetic Kerr electro-optic data obtained for an axisymmetric poidplane electrode geometry in transformer oil with specified space charge density distributions. The impact of experimental error is analyzed by incorporating random error to the synthetic data. Regularization techniques that decrease the impact of experimental error are applied. In principle FEBKER is applicable to arbitrary three-dimensional geometries as well.
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