diff --git a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex index 30dbe3c..9ae5e3f 100644 --- a/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex +++ b/roam/data/d6/e44fbc-975b-463d-87f9-878d50758b55/index.tex @@ -1010,7 +1010,7 @@ With this, we have for \(m=\pm,-N,-N+1,\ldots,-1,1,2,\ldots,N\) \label{eq:94} \begin{aligned} ω^{0}_{m\neq \pm} &= ω_{n}^{B} & ω^{0}_{\pm} &= \pm δ \cos(ψ)\\ - η^{0}_{m\neq\pm} &= η_{B} = η_S+2δΔ & η^{0}_{\pm} &= n_{S} + δΔ = + η^{0}_{m\neq\pm} &= η_{B} = η_S+2δΔ & η^{0}_{\pm} &= η_{S} + δΔ = \frac{η_{S}+ η_{B}}{2}. \end{aligned} \end{equation} @@ -1130,9 +1130,11 @@ Using \cref{sec:effects-asymm-damp}, we identify which yields using \cref{eq:88} \begin{equation} \label{eq:98} - η_{m} = \frac{\abs{κ} πn_{B}}{2c\cos(ψ)} \pqty{δ_{m,+} - \eu^{\iu ψ} + δ_{m,-}\eu^{-\iu ψ}}. + η_{m} = \frac{\abs{κ} πn_{T}}{2c\cos(ψ)} \pqty{δ_{m,+} + \eu^{\iu ψ} + δ_{m,-}\eu^{-\iu ψ}}, \end{equation} +where \(n_{T}\) is the refractive index of the transmission line +(replace \(n_{B}\to n_{T}\) in \cref{sec:micr-deriv}). To obtain the \(ω_{γ}\), \(λ_{γ}\) and \(O_{m,γ}\) we have to diagonalize \begin{equation} @@ -1146,13 +1148,13 @@ enter the final interaction hamiltonian. As \begin{equation} \label{eq:100} - \Im η_{\pm}= \pm \frac{\abs{κ} πn_{B}}{2c\cos(ψ)}Δ, + \Im η_{\pm}= \pm \frac{\abs{κ} πn_{T}}{2c\cos(ψ)}Δ, \end{equation} we obtain a correction to the on-site energies \(ε_{\pm}=\pm ε_{A}\), whereas \begin{equation} \label{eq:101} - \Re η_{\pm} = \frac{\abs{κ} πn_{B}}{2c} + \Re η_{\pm} = \frac{\abs{κ} πn_{T}}{2c} \end{equation} is independent of \(Δ\) which is consistent with the symmetry of \(η^{0}_{\pm}\). Note however that \cref{eq:100} is of very small @@ -1165,7 +1167,7 @@ In light of \cref{eq:113} we may choose\footnote{The right hand sides \begin{gather} \label{eq:114} ε_{+} = ε_{A}- \Im η_{+}= ε_{A} - \frac{\abs{κ} - πn_{B}}{2c\cos(ψ)}Δ\\ + πn_{T}}{2c\cos(ψ)}Δ\\ \begin{aligned} ε_{j} &= ω_{j}, & \varphi_{j} &= 0, & \hat{f}_{i}&= \frac{\sqrt{2}}{\hat{J}_{0,i}}η_{j} \end{aligned} @@ -1194,7 +1196,7 @@ with \label{eq:117} \tilde{ω}_{\pm} \equiv ω_{σ}^{0}-ε_{σ} = ω_{S} \pm δ\cos(ψ) - ε_{A} + \frac{\abs{κ} - πn_{B}}{2c\cos(ψ)}Δ. + πn_{T}}{2c\cos(ψ)}Δ. \end{equation} Finally we arrive at @@ -1305,7 +1307,7 @@ works out to be \begin{equation} \label{eq:71} \begin{aligned} - η_{\pm} = \frac{\abs{κ} πn_{B}}{2c\cos(ψ)} \eu^{\mp \iu ψ} + η_{\pm} = \frac{\abs{κ} πn_{T}}{2c\cos(ψ)} \eu^{\mp \iu ψ} \end{aligned} \end{equation} where the sign in the exponent is \emph{inverted} compared to