Refractive indices of biaxial crystals evaluated from the refractive indices ellipsoid equation

The expressions of the refractive indices corresponding wave vector k(θ,ϕ)k(θ,ϕ) are given by using coordinate transformations for the refractive indices ellipsoid equation of biaxial crystals. The refractive index expression of the wave, whose velocity is slower, is given by1n1‴2=[(cos2ϕn12+sin2ϕn22)cos2θ+sin2θn32]cos2δ+(sin2ϕn12+cos2ϕn22)sin2δ-12(1n12-1n22)sin(2ϕ)cosθsin(2δ),and that of the wave, whose velocity is faster, is given by1n2‴2=[(cos2ϕn12+sin2ϕn22)cos2θ+sin2θn32]sin2δ+(sin2ϕn12+cos2ϕn22)cos2δ+12(1n12-1n22)sin(2ϕ)cosθsin(2δ).tg(2δ)=-(1n12-1n22)sin(2ϕ)cosθ(cos2ϕn12+sin2ϕn22)+[(sin2ϕn12+cos2ϕn22)cos2θ+sin2θn32].