\documentclass[12pt, a4paper]{article}

\title{Dipole emission near a planar multilayer stack}
\author{Baptiste Augui\'e}    
\date{\today}
\usepackage{siunitx}
\usepackage{amsmath,amssymb}% AMS standards    
\usepackage[utf8]{inputenc}
\SweaveOpts{pdf=TRUE, echo=FALSE, fig=FALSE, eps=FALSE, cache=FALSE, tidy=T}
% \VignetteIndexEntry{planar}
\bibliographystyle{alpha}

\begin{document}

\setkeys{Gin}{width=0.6\textwidth}

\newcommand{\dq}{\text{d}q} 
\newcommand{\du}{\text{d}u} 

\maketitle
\begin{abstract}
The \texttt{planar} package solves the electromagnetic problem of dipole emission near a planar multilayer stack. It comprises two sets of functions; i) to compute the effective Fresnel reflection coefficient of a multilayer structure; ii) to evaluate the modified dipolar field as an integral over plane waves reflected at the interface.
\end{abstract}

<<setup,cache=FALSE>>=
library(planar)
library(ggplot2)
library(lattice)
library(gridExtra)
theme_set(theme_minimal())
pal1 <- grey(seq(0,1,leng=100))
## ls("package:cda")
options(demo.ask=FALSE)
@

<<dielectric_functions>>=
wvl <- seq(200, 1000,by=2)
gold <- planar:::epsAu(wvl)
silver <- planar:::epsAg(wvl)
@ 

\section{Fresnel coefficients}

The functions \texttt{recursive.fresnel} and \texttt{multilayer} both
compute the Fresnel coefficients for a multilayer stack, using two
different methods (recursive application of Fresnel coefficients for a
layer; and transfer matrix, respectively).

\subsection{Multilayer optics}

<<sbragg_stack, cache=TRUE>>=
demo(bragg_stack, echo=FALSE)
@
\begin{figure}[!ht]
\centering
<<bragg_stack, fig=TRUE>>=
print(p)
@
\caption{\texttt{demo(bragg\_stack)} from the planar package. Reflectivity of a Bragg stack with varying number of layers. Reproducing Fig. 6.6, p. 188 of Mac Leod's Thin Film Optical Filters the structure is a stack of lambda/4 layers of indices nH and nL on a glass substrate with increasing number of layers, the reflectivity stop-band becomes stronger.}
\label{fig:braggstack}
\end{figure}


<<sfield_enhancement, cache=TRUE>>=
demo(field_enhancement, echo=FALSE)
@
\begin{figure}[!ht]
\centering
<<field_enhancement, fig=TRUE>>=
print(p)
@
\caption{\texttt{demo(field\_enhancement)} from the planar package. Comparison of the calculation of near field enhancement outside of a thin metal film with Fresnel reflection and transmission coefficients.}
\label{fig:fieldenhancement}
\end{figure}

<<sLFIEF_distance, cache=TRUE>>=
demo(LFIEF_distance,  echo=FALSE)
@

\begin{figure}[!ht]
\centering
<<LFIEF_distance, fig=TRUE>>=
print(p)
@
\caption{\texttt{demo(LFIEF\_distance)} from the planar package. Local field enhancement factors for a dipole near or inside a multilayer. Note that the field and its derivative are continuous across all interfaces.}
\label{fig:LFIEFdistance}
\end{figure}

\pagebreak
\subsection{Kretschmann configuration -- planar surface plasmon-polaritons}

First, we look at the reflectivity of a thin metal film excited in the
Kretschmann configuration.
<<skretschmann_angle_scan, cache=TRUE>>=
demo(kretschmann_angle_scan,echo=FALSE)
@

\begin{figure}[!ht]
\centering
<<kretschmann_angle_scan, fig=TRUE>>=
print(p)
@ 
\caption{Reflectivity of a thin metal film, \SI{50}{nm} thick, sandwiched between glass ($n=1.5$) and air. The SPP is excited at the metal/air interface. By changing the incident angle, the normalised in-plane wavevector $q$ varies from $0$ (normal incidence) to $1$ (grazing internal angle).}
\label{fig:kretschmann_angle_scan}
\end{figure}

In the same configuration, SPPs may be excited for a wide range of
frequencies. The dispersion of the surface mode may be observed as a
high reflectivity trace when plotted as a function of incident
in-plane wavevector and energy.

<<sdispersion_kretschmann, cache=TRUE>>=
demo(dispersion_kretschmann,  echo=FALSE)
@ 
\begin{figure}[!ht]
\centering
<<dispersion_kretschmann, fig=TRUE>>=
print(p)
@ 
\caption{Reflectivity of a thin metal film, \SI{50}{nm} thick, sandwiched between semi-infinite glass ($n=1.5$) and air. The dispersion of the SPP mode appears as a dark curve following the equation $k_\text{spp} = k_0 \sqrt{\frac{\varepsilon_\text{metal}\varepsilon_\text{air} }{\varepsilon_\text{metal}+\varepsilon_\text{air} }}$}
\label{fig:dispersion_kretschmann}
\end{figure}

Free-space radiation cannot directly couple to SPP modes due to a
momentum mismatch. Using evanescent illumination, in-plane wavevectors
of arbitrarily large value may be obtained and allow the mapping of
the coupled-SPPs dispersion in a symmetric configuration.
<<sdispersion_symmetric,  cache=TRUE>>=
demo(dispersion_symmetric, echo=FALSE)
@

\begin{figure}[!ht]
\centering
<<dispersion_symmetric,fig=TRUE>>=
print(p)
@
\caption{Reflectivity of a thin metal film, \SI{50}{nm} thick, sandwiched between semi-infinite glass ($n=1.5$) on either side. Coupled SPPs are excited when the normalized in-plane wavevector $q$ is greater than 1. Note that values of $|r|^2> 1$ are not unphysical, as no power is transferred by evanescent waves.}
\label{fig:dispersion_symmetric}
\end{figure}

\clearpage

\section{Decay rates}

From \cite{RE09} (p. 571), and \cite{NH06} (pp. 335--360), the total decay rate for a dipole perpendicular to the interface is

\begin{equation}
  M^\perp_\text{tot} = 1 + \frac 3 2 \int_0^\infty \Re\left\{ \frac{q^3}{\sqrt{1 - q^2}} r^p (q) \exp\left(2 i k_1 d\sqrt{1 - q^2} \right)\right\}\dq 
  \label{eq:Mptot1}
\end{equation}

The integrand diverges as $q\to 1$, it is therefore advantageous to perform the substitution $u:=\sqrt{1 - q^2}$. In order to maintain a real path of integration, the integral is first split into a radiative region ($0\leq q\leq 1$, $u:=\sqrt{1 - q^2}\geq 0$), and an evanescent region ($1\leq q\leq \infty$, $-i u:=\sqrt{q^2 - 1}\geq 0$). After some algebraic manipulation, we obtain,
\begin{equation}
  M^\perp_\text{tot} = 1 + \frac 3 2 \left(I_1 + I_2\right)
  \label{eq:Mptot2}
\end{equation}
where
\begin{equation}
  \begin{split}
I_1 + I_2 =& \int_0^1 \left[1 - u^2\right]\cdot\Re\left\{ r^p (\sqrt{1 - u^2}) \exp\left(2 id k_1 u\right)\right\}\du \\ & +\int_0^\infty \left[1 + u^2\right]\cdot\exp\left(-2d k_1  u\right)\cdot\Im\left\{ r^p (\sqrt{1 + u^2}) \right\}\du 
  \end{split}
  \label{eq:MptotI12}
\end{equation}

Similarly, for the parallel dipole
\begin{equation}
  M^\parallel_\text{tot} = 1 + \frac 3 4 \int_0^\infty \Re\left\{ \left[ \frac{r^s(q)}{\sqrt{1 - q^2}} - r^p (q) \sqrt{1 - q^2}\right] \cdot q\cdot\exp\left(2 i k_1 d\sqrt{1 - q^2} \right)\right\}\dq  \label{eq:Mstot1}
\end{equation}
which can be rewritten as,
\begin{equation}
  M^\parallel_\text{tot} = 1 + \frac 3 4 \left(I^\parallel_1 + I^\parallel_2\right)
  \label{eq:Mstot2}
\end{equation}
where
\begin{equation}
  \begin{split}
I^\parallel_1 + I^\parallel_2 =& \int_0^1 \Re\left\{\left[r^s (\sqrt{1 - u^2})  - u^2\cdot r^p (\sqrt{1 - u^2}) \right] \exp\left(2 id k_1 u\right)\right\}\du \\ & 
+\int_0^\infty \exp\left(-2d k_1  u\right)\cdot \Im\left\{r^s (\sqrt{1 + u^2}) + u^2\cdot r^p (\sqrt{1 + u^2}) \right\}\du 
  \end{split}
  \label{eq:MstotI12}
\end{equation}

<<sdipole_integrand, cache=TRUE>>=
demo(dipole_integrand, echo=FALSE)
@
\begin{figure}[!ht]
\centering
<<dipole_integrand_map, fig=TRUE>>=
print(p)
@
\caption{Integrand in the resonance region of the total decay rate enhancement factor $M_\text{tot}$ for a dipole situated \SI{5}{nm} above a metal interface.}
\label{fig:dipole_integrand}
\end{figure}


\begin{figure}[!ht]
\centering
<<dipole_integrand_slices,fig=TRUE>>=
print(p2)
@
\caption{Integrand of the total decay rate enhancement factor $M_\text{tot}$ for a dipole situated \SI{5}{nm} above a metal interface, for several emission wavelengths.}
\label{fig:dipole_integrand2}
\end{figure}

<<sEM_enhancements, cache=TRUE>>=
demo(EM_enhancements, echo=FALSE)
@
\begin{figure}[!ht]
\centering
<<EM_enhancements,fig=TRUE>>=
print(p)
@
\caption{Total and radiative decay rate enhancements for a dipole near a metal interface. Reproducing Fig. 6.1, p. 304 from Principles of Surface-Enhanced Raman Spectroscopy. A dipole is placed near a semi-infinite air/metal interface with orientation either parallel or perpendicular to the interface the total decay rates peak at the wavelength of excitation of planar SPPs epsilon=-1 at the interface (loss channel). The radiative decay rate in the upper medium has a trough at the wavelength where $\varepsilon=0$ ($Dn=0$, by continuity $En=0$).}
\label{fig:EM_enhancements}
\end{figure}

<<sintegrated_fluorescence,  cache=TRUE>>=
demo(integrated_fluorescence, echo=FALSE)
@
\begin{figure}[!ht]
\centering
<<integrated_fluorescence,fig=TRUE>>=
print(p)
@
\caption{Integrated decay rates and efficiency for a dipole near a semi-infinite air/metal interface
 for gold and silver, varying the wavelength and the dipole-interface distance.}
\label{fig:integrated_fluorescence}
\end{figure}


<<skretschmann_fluo_distance, cache=TRUE>>=
demo(kretschmann_fluo_distance, echo=FALSE)
@ 
            

\begin{figure}[!ht]
\centering
<<kretschmann_fluo_distance,fig=TRUE>>=
print(p)
@
\caption{Fluorescence enhancement vs distance in the Kretschmann configuration.}
\label{fig:kretschmann_fluo_distance}
\end{figure}


\subsection{Angular pattern of dipole emission}
By virtue of reciprocity, the local field intensity enhancement factor also represent the probability of emission of a dipole in a particular direction. 


<<sfluorescence_pattern,  cache=TRUE>>=
demo(fluorescence_pattern, echo=FALSE)
@
\begin{figure}[!ht]
\centering
<<fluorescence_pattern,fig=TRUE>>=
grid.arrange(p1, p2, p3, p4, ncol=2)
@
\caption{Radiation pattern of a dipole near a dielectric/(metal)/dielectric interface parallel and perpendicular orientations, p- and s- polarisations.}
\label{fig:fluorescence_pattern}
\end{figure}

\bibliography{refs}

\end{document}

