Transform Relationship between Kerr - effect Optical Phase Shift and Nonuniform Electric Field Distributions Markus

Electricfield distributions measured using the Kerr effect cause a phase,shift between light components polarized parallel and perpendicular to the electric field, proportional to the magnitude sqyared of the electric field components in the plane perpendicdar to light propagation integrated over the light path length. One wishes to recover the electric field distribution from m4asurements of light phase shifts. For axisymmetric geometriet where the electric field depends only on the radial coordinate and whose direction is constant along the light path, as is the case along a planar electrode, the total phase shift for lighQ propagating at a constant distance from the center of symmewy and the electric field distribution are related by an Abel trpsform pair, a special case of Radon transforms typically used in image reconstructions with medical tomography and holography. The more general Radon transform relates the optical phase shift to non-axisymmetric electric field distributions bu;t is restricted to cases where the applied electric field is perpenc&ular to the plane of light propagation. If the applied electric peld direction changes along the light path, it becomes necessaq to account for the change in direction of the light compon nts parallel and perpendicular to the applied electric field an a the light polarization equations are generalised. 1. INTRODWCTION 1.1. BACKGROUND ATHEMATICAL and phyrical analysis of the scalar M potential and electric field pistributions in high field environments usually take insullrting dielectrics to be uncharged, both in the volume and on the surface. In practice, this is usually not true becquse conduction results in surface charge on interfaces and charge injection results in volume charge. The potenti4 and electric field distributions cannot then be calculatd from knowledge of ayetem geometries and material dlelectric properties alone using Laplace’s equation, aa the system is then described by Poisson’s equation and the conduction laws must be known to calculate the surface and volume charge distributions. Since charge injection and transport are related to the electric field which in turn depends on the charge distribution via G a u d law, the electric field and charge distribution must be self-consistently determined. 1.2. SCOPE Since the conduction laws are often unknown, it is desired to determine the magnitude and direction of an 1070-9878/94/ 83.00 @ 1994 IEEE Authorized licensed use limited to: MIT Libraries. Downloaded on February 23, 2009 at 15:56 from IEEE Xplore. Restrictions apply. 236 Zehn: Kerr-effect Optical Phase Shift and Nonuniform Electric Field Distributions arbitrary applied electric field distribution using electric field induced birefringence [Kerr effect) measurements. For electric field distributions that have constant direction along the optical path length, even though spatially varying in magnitude, this work shows that the measured phase shift is related to an axisymmetric electric field distribution by an Abel transformation, and to a non-axisymmetric electric field distribution that is perpendicular to the plane of light propagation by a Radon transformation. A few special cases with constant direction electric fields are presented. However, most problems of interest do not have a constant electric field direction along an optical path. Then, the optical components parallel and perpendicular to the applied electric field direction also change direction as the applied electric field direction changes. An exact special caae with constant magnitude electric field but changing direction and an approximate solution when the electric field spatial variations are very gradual compared to the optical wavelength are given for the light output through linear and circular polariscopes. &om a measured optical intensity distribution, continued research is necessary to totally recover information on the magnitude and direction everywhere of an appl&d nonuniform electric field i z Figure 1. If the applied electric field 81 in the plane perpendicular to the z direction of light propQation varies in magnitude and direction along the optical path, the com_ponents of light electric field parallel (Zll = ellill) and perpendicular (CL = e l l l ) to the applied electric field change both in phase and direction. In general, the direction of electric field bl at angle 8, to the 1: axis in the t y plane can be a function of z. constant direction, it is useful to rederive the Kerr effect distribution. 2. KERR ELECTRO-OPTIC FIELD relations, without any simplifyingassumptions about the applied electric field. We take the light to be z directed so that only the z MAPPING MEASUREMENTS 2.1. LIGHT POLARIZATION IN NONUNIFORM ELECTRIC FIELDS In order to determine the charge injection and transport physics, many researchers use the Kerr electro-optic effect to measure electric fiew distributions, whereby the changes in the refractive indices for light polarized parallel (rill) and perpendicular (nl) to an applied electric field are proportional to the magnitude squared of the electric field components, ( I& l 1 2 ) , in the plane perpendicular to the direction of light propagation [l] and y components of the applied electric field have any birefringent effects: We de_note the electric field vector of the light as e'= erae + eu& and take the applied electric field in the zy plane to be of the form 2~ = Eziz + Eu& (3) where the electric field components c_an xary wiih coordinates x , y and z, with unit vectors i , , iu, and a,. Any z component of applied electric field has no birefringent effect on light propagating in the z direction. The unit vectors parallel and perpendicular to the applied electric field are then Ezi, + E#& where A is the free space optical wavelength and B is the Kerr constant. There is no change in refractive index due to applied electric field components along the direction of light propagation. In terms of wavenumbers parallel and perpendicular to the applied electric field, this becomes