Total and radiative decay rate modifications for a dipole near a semi-infinite metal interface

Setting up

wvl <- seq(200, 1000,by=2)
silver <- epsAg(wvl)
gold <- epsAu(wvl)

distance <- function(d=1, material="silver", ...){

  material <- get(material)
  
  dl <- dipole(d=d,
               wavelength = material$wavelength,
               epsilon = list(incident=1.0^2, material$epsilon),
               thickness = c(0, 0),
               Nquadrature1 = 1e3, Nquadrature2 = 5e3, GL = FALSE,
               Nquadrature3 = 5e3, qcut = NULL, rel.err=1e-3,
               show.messages = FALSE)

  message(attr(dl, "comment"))
  
  m <- melt(dl, id = "wavelength")

  m$orientation <- m$variable
  
  levels(m$orientation) <- list(perpendicular="Mtot.perp",
                                perpendicular="Mrad.perp",
                                parallel="Mtot.par",
                                parallel="Mrad.par")
  
  levels(m$variable) <- list(Mtot="Mtot.perp",
                             Mtot="Mtot.par",
                             Mrad="Mrad.perp",
                             Mrad="Mrad.par")
  invisible(m)
}

Reproducing Fig. 6.1, p. 304 from Principles of Surface-Enhanced Raman Spectroscopy.

params <- expand.grid(d=c(1,5,10), material=c("silver", "gold"), 
                      stringsAsFactors = FALSE)
all <- mdply(params, distance)

## relative integration errors were: 4.737e-05 for I1,  1.445e-03 for I2,  3.957e-01 for I3,  4.746e-04 for I4; with 195, 1455, 285, 165 respective function evaluations.

## relative integration errors were: 2.144e-05 for I1,  1.397e-03 for I2,  3.314e-04 for I3,  5.414e-04 for I4; with 225, 1455, 225, 165 respective function evaluations.

## relative integration errors were: 2.004e-05 for I1,  1.338e-03 for I2,  3.813e-05 for I3,  6.288e-04 for I4; with 225, 1455, 195, 165 respective function evaluations.

## relative integration errors were: 1.124e-04 for I1,  1.079e-03 for I2,  4.955e-01 for I3,  6.125e-06 for I4; with 45, 495, 285, 75 respective function evaluations.

## relative integration errors were: 1.115e-04 for I1,  1.068e-03 for I2,  5.417e-03 for I3,  6.078e-06 for I4; with 45, 495, 225, 75 respective function evaluations.

## relative integration errors were: 1.104e-04 for I1,  1.055e-03 for I2,  5.094e-03 for I3,  6.020e-06 for I4; with 45, 495, 195, 75 respective function evaluations.

ggplot(all, aes(wavelength, value, colour=factor(d), linetype=orientation))+
  facet_grid(variable~material, scales="free_y") + 
  geom_path() + labs(colour="distance /nm", y="EM enhancement factor", x="wavelength /nm")+
  scale_y_log10() 

The total decay rates peak at the wavelength of excitation of planar SPPs (ε = −1) at the interface, they represent a non-radiative loss channel. The radiative decay rate in the upper medium has a trough at the wavelength where ε = 0 (D = 0 in the metal, by continuity En = 0 above the interface), therefore by virtue of reciprocity the perpendicular dipole will not emit.