# Optical mode leakage through a layer of gold

The geometry of my semiconductor device is given below. The blue regions are gold, the grey ones - gallium arsenide (n-doped to $2.9 \times 10^{15} \mathrm{cm^{-3}}$). The dimensions are μm, i.e. it is 500μm wide, the bottom GaAs is 400μm thick, then a layer of gold follows, then a 15μm layer of GaAs, and a 5μm layer of Au on the top:

The thinner strip of GaAs is a laser medium where radiation of 3THz is emitted. I would like to investigate the effect of the thickness of the thin layer of gold on the optical mode profile. I simulated the profile for two thicknesses: 20nm and 200nm. Below are the plots of the normalised E field, first 2D and then a cross section along x=0 line. Arc length is simply the y axis of the first plot.

20nm

200nm

In the first (20nm) case, |E| decreases from 450V/m to 90V/m (5x). In the second (200nm) case, |E| decreases from 2800V/m to 10V/m (280x).

The properties of the materials are as follows (Drude model for GaAs, literature for Au):

+----------+--------+---------+
| Material | Im(ε)  | Re(ε)   |
+----------+--------+---------+
| Au       | 1.43e5 | -5.11e4 |
| GaAs     | -0.17  | 12.98   |
+----------+--------+---------+


The field should decrease as $\exp(-\alpha z)$ (Beer's law), where $\frac{1}{\alpha} = \delta \approx 50\mathrm{nm}$ is the skin depth for 3THz in gold - this is calculated from: $$\alpha = \frac{1}{\delta} = \frac{\kappa \omega}{c}$$ where $\kappa$ is the known value for $Im(\tilde{n}) = 319$ (Ordal et al., http://dx.doi.org/10.1364/AO.26.000744). Hence: $$E_2(20nm) = 450 \exp(-20/50) \approx 300$$ $$E_2(200nm) = 2800 \exp(-200/50) \approx 50$$

The first result is 3x higher and the second 5x higher than I see in the simulation.

To check this, I simulated the system for different thicknesses of the gold layer, and fitted the data to the equation: $$\frac{|E_2|}{|E_1|} = A \exp \left(-\frac{z}{\delta} \right)$$

I got the following plot:

And the fit yields these parameters: $$\frac{|E_2|}{|E_1|} = 0.32 \exp \left(-\frac{z}{41\mathrm{nm}} \right)$$

So $\delta=41\mathrm{nm}$ is not too far from the calculated value. What I am trying to understand now is:

Where does the factor of 0.32 come from? It obviously matters much more for lower thickness of gold, which I am interested in. Does it have something to do with the normalisation? Boundary conditions?

I have realised that I should be looking at $\mathrm{E_y}$ rather than $\mathrm{|E|}$, but I checked it and the values look the same.

COMSOL FEM software was used to visualise the problem below. The question is related - albeit different - to this one: The skin effect and the reflectivity of gold

• what is a mode in this case? Do you have Dirichlet boundary conditions somewhere or does the software simulate some leaky situation? – Wolpertinger Apr 6 '17 at 9:38
• All external boundaries are scattering (absorbing). – alkamid Apr 6 '17 at 10:29

• What's the value that you got? I double checked and $\kappa = 319$, both on refractiveindex.info and in Ordal's paper. – alkamid Mar 17 '16 at 18:28
• You are using $Im(n)$ to determine the skin depth. Are you also using it to determine optical losses in the Au layer? In your table, you list an imaginary value of 0 for Au. – engineer Mar 21 '16 at 18:14
• Sorry, that 0 in the table was confusing — I was using a loss model in COMSOL which only required one part of $\epsilon$ and conductivity, as they are dependent. – alkamid Apr 6 '17 at 8:57