6 B. Magnetic moment The magnetization in gyrotropic media is Taking solution for the circularly polarized wave we come to the dispersion relation It is worth to be noted that for a media with time reversal breaking one has to substitute here σ → σ$_{±}$ = σ$_{xx}$ ± iσ$_{xy}$ , where σ$_{xy}$ is the Hall conductivity. In neglect the gyrotropy term the complex index of refraction such that rotation of the magnetization is equal to one half of gyrotropy part of the current density The relationship between the density of gyrotropy cur- rent and the magnetization is a general property of non- centrosymmetric materials (see also $^{13}$). Both of them can be obtained from the gyrotropy term in action is expressed through the diagonal part of complex con- ductivity σ = σ ′ + iσ ′′ by means of the usual relations M = 1 2 c λ E , (62) E = (ˆ x ± i ˆ y ) E$_{0}$e i ( kr − ωt ) (55) k 2 = ϵω 2 c 2 + 4 πiσω c 2 ± 4 πiλωk c 2 . (56) 1 2 j$_{g}$ = c rot M . (63) − 1 2 c ∫ dtd $^{3}$r ( λ E ) B . N = ck ω = n + iκ n 2 − κ 2 = ϵ − 4 πσ ′′ ω , 2 nκ = 4 πσ ′ ω . N ± = n + iκ ± 2 πiλ c . (57) S$_{g}$ = − 1 2 c ∫ dtd $^{3}$r { ( λ E ) B + ν [ A − ¯ hc 2 e ∇ ϕ ] B } . (64) M = 1 2 c λ E + 1 2 c ν [ A − ¯ hc 2 e ∇ ϕ ] , (65) Δ n = n$_{+}$ − n$_{−}$ = − 4 πλ ′′ c , (58) Δ κ = κ$_{+}$ − κ$_{−}$ = 4 πλ ′ c . (59) By variation of action in respect of − B and − A /c , taking into account that λ is an odd function of derivative ∂/∂t and making use the definitions E = − (1 /c ) ∂ A /∂t and B = rot A , we come to M given by eqn.(62) and j$_{g}$ given by eqn.(53) correspondingly. All these considerations are valid both for the normal and as well for the superconducting state. However, in the latter case the gyrotropy action The gyrotropy term leads to the difference in the refrac- tion indices of clock wise and counter clock wise polarized light. In the first order in respect to λ = λ ′ + iλ ′′ the refraction index is Hence, the differences in the real and imaginary parts of the refraction indices of circularly polarized lights with the opposite polarization are contains one extra term which is absent in the normal state due to the gauge invariance. The corresponding expressions for the magnetic moment and gyrotropy cur- rent are In the superconducting state the gyrotropy current (53) has more general form given by eqn. (4). Hence, we should use the more general formula for the current Then repeating all the calculations we come to the same results (56)-(59) modified by the substitution To consider the problem of light reflection normally incident to the flat surface of noncentrosymmetric metal we need to find the relations between the wave amplitude propagating inside ( z > 0) the material C. Boundary conditions j$_{g}$ = λ rot E + ν B . (66) j = ϵ 4 π ∂ E ∂t + σ E + λ rot E + ν B . (60) λ → Λ = λ − icν ω . (61) E in = E$_{0}$ e iω ( Nz/c − t ) (67) E out = E$_{1}$ e iω ( z/c − t ) + E$_{2}$ e − iω ( z/c + t $^{)}$. (68) E in z = o = E out z =0 and the amplitudes of incident and reflected waves out- side it We have We remind that the superconducting state current den- sity given by eqn. (60) is worth to use at the high frequencies ω > v$_{F}$ /δ where the inequality ¯ hω >> Δ is certainly valid. Here, Δ is superconducting energy gap. In the low frequency limit ¯ hω < Δ one should also take into account the London density of current j$_{L}$ = − ( c/ 4 πδ 2 $_{L}$)( A − ¯ hc ∇ ϕ/ 2 e ). The interplay between the London current j$_{L}$ and the Drude current j$_{D}$ = σ ( ω ) E is discussed in the textbook. 16