lecture17

# lecture17 - 1 ECE 303 – Fall 2007 – Farhan Rana –...

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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 ε ε = + = Now consider a material made up molecules that can easily be polarized by E-fields in the z-direction, but don’t 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 . ε = 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 ε ε ε ε ε ε ε ε ε • 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 ε ε = ECE 303 – Fall 2007 – Farhan Rana – Cornell University Biaxial Media The dielectric permittivity matrix is symmetric, i.e. ( ) T ε ε = 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 ε ε ε • 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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lecture17 - 1 ECE 303 – Fall 2007 – Farhan Rana –...

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