Slow light in photonic crystals by Alex Figotin; Ilya Vitebskiy
By Alex Figotin; Ilya Vitebskiy
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In the case n = 4 there is no W (0), and in the case n = 2 in a generic situation W (0) will be just a diagonal matrix. Slow light in photonic crystals 56 Consequently, the basis fj , j = 0, 1, 2, 3 reducing T (0) to the above mentioned Jordan form (274), (275) can represented as follows fj = G0 (0) bj , j = 0, 1, 2, 3 where 0 0 0 1 0 0 1 0 b0 = 0 , b1 = 0 , b2 = 1 , b3 = 0 1 0 0 0 (276) . 8. Spectral perturbation theory of the transfer matrix at a point of degeneracy In this section we develop the spectral perturbation theory for the transfer matrix T (ω) defined by (101).
238) In particular, an inflection point k0 is a 3-fold degenerate point if ω ′ (k0 ) = ω′′ (k0 ) = 0, ω ′′′ (k0 ) = 0. (239) Hence, if k0 is a n-degenerate point we have ω (k) = ω (k0 ) + ∂kn ω (k0 ) n n+1 , k → k0 . (k − k0 ) + O (k − k0 ) n! (240) In particular, if k0 is an inflection point then ω (k) = ω (k0 ) + ω ′′′ (k0 ) 3 4 (k − k0 ) + O (k − k0 ) , k → k0 . 6 (241) To study the behavior of the transfer matrix TL near ω 0 we introduce T (ν) = T (ω 0 + ν) , ν = ω − ω 0 . (242) Slow light in photonic crystals 52 We assume the dependence of T (ν) on ν to be analytic in some vicinity of ν = 0.
2 k32 + k12 k22 Notice also that (v+ , j2 v− ) = (v− , j2 v+ ) = 0, (v± , j2 v± ) = ±i, (v± , v± ) = β ω,kτ = (v∓ , v± ) = k32 + 2 k22 γ 2ω,kτ (168) 2 + (ωk3 ) + (k1 k2 ) , 2 (k32 + k22 ) k3 ω k32 + k22 ±iωk3 + k1 k2 1 2 2 +1 , (u± , u± ) = 1, (u∓ , u± ) = 0; (u∓ , j2 u± ) = 0, (u± , j2 u± ) = ±i. (0) Using the tensor product representation (163) for JA we obtain (169) and (166) and (167), (168) JA(0) Z1± = ±k3 Z1± , JA(0) Z2± = ±k3 Z2± , Zj± = Zj± (ω, kτ ) . (170) Z1+ = u− ⊗ v+ , Z2+ = u+ ⊗ v− , Z1− = u+ ⊗ v+ , Z2− = u− ⊗ v− .