lecture17 - 1 ECE 303 Fall 2007 Farhan Rana Cornell...

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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 Quarter-wave and half-wave 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 D-field is related to the E-field 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 E-fields in the z-direction, but dont respond much to E-fields in the x- and y-directions, as shown in the figure z x y z x y E-field in the x- or y-directions: material not much polarized-ve electron cloud +ve ions E-field 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 D-field is related to E-field through a dielectric permittivity tensor: ( ) ( ) r E r D r r r r . = What this really means is that the x-, y-, and z-components of the D-field are related to the x-, y-, and z-components of the E-field 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 co-ordinate axes w.r.t. the material) So in the most general case, if one chooses the orientation of the co-ordinate axes judiciously, the relation between the D-field and E-field 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 co-ordinate 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).

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lecture17 - 1 ECE 303 Fall 2007 Farhan Rana Cornell...

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