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Unformatted text preview: 1 ECE 303 Fall 2007 Farhan Rana Cornell University Lecture 17 Waves in Anisotropic Media In this lecture you will learn: Wave propagation in anisotropic dielectric media Wave propagation in biaxial and uniaxial media Birefringence Quarterwave and halfwave plates ECE 303 Fall 2007 Farhan Rana Cornell University Anisotropic Media So far you have been dealing with materials that looked the same in all directions (i.e. isotropic) For isotropic media, the Dfield is related to the Efield by one number the dielectric permittivity: ( ) ( ) ( ) ( ) r E r P r E r D o r r r r r r r r = + = Now consider a material made up molecules that can easily be polarized by Efields in the zdirection, but dont respond much to Efields in the x and ydirections, as shown in the figure z x y z x y Efield in the x or ydirections: material not much polarizedve electron cloud +ve ions Efield in the z direction: material strongly polarized 2 ECE 303 Fall 2007 Farhan Rana Cornell University Dielectric Permittivity Tensor In the most general case, the Dfield is related to Efield through a dielectric permittivity tensor: ( ) ( ) r E r D r r r r . = What this really means is that the x, y, and zcomponents of the Dfield are related to the x, y, and zcomponents of the Efield by a permittivity matrix: ( ) ( ) ( ) ( ) ( ) ( ) = r E r E r E r D r D r D z y x zz zy zx yz yy yx xz xy xx z y x r r r r r r This is the most general way of representing the effects of material polarization when the material is anisotropic The permittivity matrix is always symmetric, i.e. . This follows from physical considerations that have to do with energy conservation and time reversal symmetry. No media can violate this condition ( ) T = ECE 303 Fall 2007 Farhan Rana Cornell University Biaxial Media The dielectric permittivity matrix is symmetric, i.e. ( ) T = There is a theorem in linear algebra that says that any symmetric matrix can be diagonalized by a suitable choice of the basis vectors (i.e. by a suitable choice for the orientation of the coordinate axes w.r.t. the material) So in the most general case, if one chooses the orientation of the coordinate axes judiciously, the relation between the Dfield and Efield becomes: ( ) ( ) ( ) ( ) ( ) ( ) = r E r E r E r D r D r D z y x zz yy xx z y x r r r r r r This permittivity matrix is diagonal If the diagonal entries are all different the material is called biaxial The choice of coordinate axes that results in a diagonal permittivity matrix is called the principal axes of the material 3 ECE 303 Fall 2007 Farhan Rana Cornell University...
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This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 RANA
 Electromagnet

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