# Where n ? 2 Îµ ? 29 Îµ since a ? a ? ? a c t 1 or k a

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where n λ 2 = ε λ ( 29 ε 0 . Since ) a λ ( a λ λ = a c = t 1 (or ( k ) a λ ( 29 ( a λ ) k ( 29 λ = a c = 1 ) we may also write Equation [ V-15a ] as ) k ( a a ( 29 ( k ) a a ( 29 1 n 2 - 1 n a 2 + ) k ( a b ( 29 ( k ) a b ( 29 1 n 2 - 1 n b 2 + ) k ( a c ( 29 ( k ) a c ( 29 1 n 2 - 1 n c 2 = 0 [ V-15b ] This latter expression is the famous Fresnel equation of wave normals . 20 A PPLICATIONS OF THE F ORMAL S OLUTION : Uniaxial Dielectric Crystals : For an optical material with uniaxial symmetry, the inverse dielectric tensor in the principal axes system must have the form 21 ε - 1 = ε - 1 ˆ a a ˆ a a + ˆ a b ˆ a b ( 29 + ε || - 1 ˆ a c ˆ a c . [ V-16 ] Thus, Equation [ I-19b ] becomes 1 n 2 - 1 n 2 1 n 2 - 1 n || 2 sin 2 ϑ + 1 n 2 - 1 n 2 cos 2 ϑ = 0 [ V-17 ] so that n o -2 = n -2 ; and n e -2 = n -2 cos 2 ϑ + n || -2 sin 2 ϑ [ V-18 ] 20 In its commonly used form, the Fresnel equation becomes ˆ k ( 29 x 2 1 n 2 - 1 n x 2 + ˆ k ( 29 y 2 1 n 2 - 1 n y 2 + ˆ k ( 29 z 2 1 n 2 - 1 n z 2 = 0 21 In this instance there is no need to trouble ourselves about conjugate unit vectors.

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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 47 R. Victor Jones, December 19, 2000 where ˆ k ˆ a c = cos ϑ . The subscript "o" identifies the "ordinary" mode and the subscript "e" the "extraordinary" mode. These results are usually plotted as follows: where the intersections of the v k vector yield the "ordinary" and "extraordinary" velocities of propagation for a given v k . Further, if we take ˆ k = sin ϕ sin ϑ ˆ a a + cos ϕ sin ϑ ˆ a b + cos ϑ ˆ a c [ V-19 it is a bagatelle to show that ) t (o) ( ( k ) = ( t (o) ( ) k ) = cos ϕ ˆ a a - sin ϕ ˆ a b [ V-20a ] ) t (e) ( ( k ) = ( t (e) ( ) k ) = cos ϑ cos ϕ ˆ a a + sin ϕ ˆ a b ( 29 - sin ϑ ˆ a c [ V-20b ] and that these quations are consistent with Equations [ V-10 ] and [ V-11 ].
O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 44 R. Victor Jones, December 19, 2000 k t (e) t (o) Magneto-optical Media : For a simple magneto-optical substance we may write the dielectric dyadic in the form 22 22 See, for example, Section 82 in L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media , Pergamon Press (1960).

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O N C LASSICAL E LECTROMAGNETIC F IELDS P AGE 45 R. Victor Jones, December 19, 2000 t ε - 1 = 1 + i γ ˆ a a ˆ a b - ˆ a b ˆ a a ( 29 [ ] ε - 1 [ V-21 ] If we introduce the conjugate principal axes ) a + = ( a - = 1 2 ( 29 ˆ a a + i ˆ a b ( 29 ) a - = ( a + = 1 2 ( 29 ˆ a a - i ˆ a b ( 29 ) a || = ( a || = ˆ a c [ V-22 ] we obtain the dielectric dyadic in the so called normal form -- viz. t ε - 1 = 1 ( 29 ) a + ( a + + 1 + γ ( 29 ) a - ( a - + ) a || ( a || [ ] ε - 1 [ V-23 ] Again from Equation [ V-15b ] it is trivial to show that n ± - 2 = 1 ± γ cos ϑ [ ] ε - 1 [ V-24 ] Using the resolution of ˆ k as given in Equation [ V-4 ] we may show that ) t ± ( ( k ) = i 2 exp - i ϕ ( 29 1 m cos ϑ [ ] ) a + - exp i ϕ ( 29 1 ± cos ϑ [ ] ) a - ± 2 sin ϑ ) a || { } [ V-25a ] and ( t ± ( ) k ) = i 2 - exp i ϕ ( 29 1 m cos ϑ [ ] ( a + + exp - i ϕ ( 29 1 ± cos ϑ [ ] ( a - m 2 sin ϑ ( a || { } [ V-25b ] E NERGY F LOW IN A NISOTROPIC M EDIA As previously noted, the content of
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