## Visualisation of the power loss as a function of in-plane wavevector for a dipole near a thin metal film
_baptiste Auguié -- `r format(Sys.time(), "%d %B, %Y")`_

A dipole near a thin silver film can excite SPPs (on either side of the film, if sufficiently thin), resulting in a strong increase of the total decay rate. The calculation of Mtot requires an integral over the full spectrum of plane waves -- propagating (q<1) and evanescent (q>1). Here we visualise the integrand as a function of wavelength and q. 

```{r , echo=FALSE,results='hide', message=FALSE}
library(knitr)
library(planar)
library(dielectric)
library(lattice)
library(ggplot2)
require(reshape2)
require(plyr)
opts_chunk$set(fig.path="dipoleintegrand/", 
               warning=FALSE,error=FALSE,message=FALSE,tidy=FALSE)
library(ggplot2)
theme_set(theme_minimal() + theme(panel.border=element_rect(fill=NA)))
```

#### Setting up

```{r , results='hide', fig.width=10}

integrand <- function(d=1, q, material=epsAg, wavelength=500,...){
  mat <- material(wavelength)
  int <- integrand_mtot(d=d, q=q, wavelength= mat$wavelength*1e3,
                          epsilon = list(incident=1.0^2, mat$epsilon, 1.0^2),
                          thickness = c(0, 50, 0))

  m <- data.frame(wavelength = rep(mat$wavelength*1e3, length(q)),
                  k0=2*pi/rep(mat$wavelength*1e3, length(q)),
                  q = rep(q, each=length(mat$wavelength)),
                  perpendicular=c(int$integrand.p),
                  parallel=c(int$integrand.s))

  melt(m, id=1:3)
}

## focus on the non-radiative and SPP region at large kx
spps <- integrand(q=seq(0,2000, length=300), wavelength=2*pi/seq(0.006, 0.03, length=300))

ggplot(spps, aes(q, k0, fill=value))+
  geom_raster() + facet_wrap(~variable) +
  labs(x = expression(q==k[x] / k[1]), y=expression(k[0]), fill="Integrand")+
  scale_x_continuous(expand=c(0,0))+
  scale_y_continuous(expand=c(0,0)) + 
  theme_minimal() + 
  theme(panel.background=element_rect(fill=NA))

## focus on the dipole-image region at very large kx
dimage <- integrand(q=seq(0,3e8, length=300), wavelength=2*pi/2.0e-5)

ggplot(dimage, aes(q, value))+
  geom_line() + geom_blank(aes(y=value*1.1)) +
  labs(x = expression(q==k[x] / k[1]), y="Integrand")+
  scale_x_log10(expand=c(0,0))+
  scale_y_continuous(expand=c(0,0)) + 
  theme_minimal() + 
  theme(panel.background=element_rect(fill=NA))

```

At very large q, the system is well-described by quasi-static image dipole interaction with a slow decay. This results in a time-consuming integration for Mtot, unless special care is taken to transform the integrand; in `dipole()`, the following transformation is used, $$\int_a^\infty f(x)dx = \int_0^1 f(a + t/(1-t)). 1 / (1-t)^2 dt$$
