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ls1_unit_5 - ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 43 V...

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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 43 R. Victor Jones, December 19, 2000 V. P LANE W AVE P ROPAGATION IN A L INEAR , H OMOGENEOUS , A NISOTROPIC D IELECTRIC M EDIA (“C RYSTAL O PTICS ”): Our objective here is to formulate a general approach to the subject of wave propagation in anisotropic dielectrics which makes use of ideas familiar from other branches of mathematical physics -- viz. the “eigenvalue problem.”. 18 For reasons that will soon become abundantly clear, treatments of “crystal optics” focus on the behavior of the dielectric displacement vector, r D ( r r , ϖ ) rather than on the electric field vector. 19 For non- magnetic dielectrics the components of the dielectric displacement are usefully represented as the Cartesian coordinates coordinates of figure called the “ellipsoid of wave normals,” the “optical indicatix,” the “index ellipsoid” or the “reciprocal ellipsoid.” 18 Kaiser S. Kunz in 1977 presented a similar treatment of this problem in a paper entitiled “Treatment of optical propagation in crystals using projection dyadics,” Am. J. Phys., Vol. 45 , 1977, pp. 267-269. 19 Perhaps the most authoritative treatment of “Crystal Optics” is found in Max Born and Emil Wolf, Principle of Optics , Pergamon Press (1986), Chapter 14. Since the stored electrical energy is given by U e = 1 2 ( 29 r D r E = 1 2 ( 29 r D t ε -1 r D we can write in the principle axis system D x 2 U e 2 ε x + D y 2 U e 2 ε y + D z 2 U e 2 ε z = 1 where ε x , ε y , ε z { } are principle axis values of the dielectric constant tensor.
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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 43 R. Victor Jones, December 19, 2000 Now let us first combine Equations [ I-8a ] and [ I-8b ] to obtain a generalized Helmholz equation for a homogeneous anisotropic dielectric r ∇× r ∇× t ε - 1 ( ϖ ) r D ( r r , ϖ ) [ ] ( 29 = ϖ 2 μ o r D ( r r , ϖ ) [ V-1 ] In general, we would hope to be able to find a set of eigenmodes of the homogeneous problem that would satisfy the scalar eigenequation 2 D ( σ ) ( r r , ϖ ) + n σ ( 29 2 k 0 2 D ( σ ) ( r r , ϖ ) = 0 [ V-2 ] where n σ ( 29 is the effective index of refraction of the σ -th eigenmode. We can be quite specific for the case of plane wave eigenmodes where we suppose that all fields have a spatial dependence exp( m i r k r r ) . From Equation [ I-8c ] we see that r D ( r r , ϖ ) must, in general, be orthogonal to r k so that we may write r D ( r k , ϖ ) = D (1) ( r k , ϖ ) ) t (1) ( ( k , ϖ ) + D (2) ( r k , ϖ ) ) t (2) ( ( k , ϖ ) [ V-3 ] where the ) t ( σ ) ( ( k )'s are polarization unit vectors which are orthogonal to ( k , the unit vector parallel to r k . Since the components of t ε in the general case may be complex -- e.g., in the case of magneto-optical media -- the eigenmodes may be polarized along "complex directions" -- e.g., “screw axes” defining the sense of circular polarization -- and we must use great care in all vector manipulations.
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