Slow light in photonic crystals by Alex Figotin; Ilya Vitebskiy

By Alex Figotin; Ilya Vitebskiy

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The J-unitarity (91) of the transfer matrix T = T (z, z0 ) imposes the following constraint on its set of four eigenvalues ζ i , i = 1, 2, 3, 4 −1 −1 −1 {ζ ∗1 , ζ ∗2 , ζ ∗3 , ζ ∗4 } ≡ ζ −1 , 1 , ζ2 , ζ3 , ζ4 (92) |det T | = 1. 1. The transfer matrix of a stack of uniform layers The greatest advantage of the transfer matrix approach stems from the fact that the transfer matrix TS of an arbitrary stack of layers is a sequential product of the transfer matrices Tm of the constituent layers TS = Tm .

183) More precisely, all possible relevant eigenmodes are described by solutions to the following Cauchy problem ∂3 Ψ (x3 ) = iJA (x3 ) Ψ (x3 ) , Ψ (a) = Φ ∈ ST (a, ω) , −∞ < x3 < ∞. (184) The two-dimensional space ST (a, ω, kτ ) provides a convenient way to describe and parametrize the relevant modes. For instance, assuming that we know ST (a, ω, kτ ) let us pick any Φ ∈ ST (ω, kτ ) and find values of the eigenmode Ψ (x3 ) in the air. The eigenmode Ψ (x3 ) can be represented as the following linear combination for −∞ < x3 < ∞ : + + + − − − −ik3 x3 Ψ (x3 ) = eik3 x3 α+ α− 1 Z1 + α2 Z2 + e 1 Z1 + α 2 Z2 , Ψ (0) = + α+ 1 Z1 + + α+ 2 Z2 + − α− 1 Z1 + − α− 2 Z2 Φ (185) ∈ ST (0) , where evidently the two pairs of coefficients α+ = α+ 1 α+ 2 and α− = α− 1 α− 2 (186) are respectively related to the incident and the reflected waves.

ST (0; ω, kτ ) = C2 , α± (Φ) = 1 β ω,kτ Z1+ , Φ Z2+ , Φ , (213) The relation (213) can be considered as another fundamental property of the space ST (0; ω, kτ ). Using the coefficients α± (Φ) and (176) we get the following representation for the flux of the mode described by Φ [Φ, Φ] = Φ+ , Φ+ − Φ− , Φ− = α+ (Φ) Φ= + α+ 1 Z1 + + α+ 2 Z2 + − α− 1 Z1 + 2 − α− 2 Z2 2 − α− (Φ) , ∈ ST (0; ω, kτ ) . (214) Slow light in photonic crystals 49 The above equality reflects the fundamental energy flux balance of the classical scattering theory in its the simplest form: α+ (Φ) 2 2 = α+ (Φ) − α− (Φ) 2 2 α− (Φ) (Incident wave flux) − (Reflected wave flux) (215) (Transmitted wave flux).

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