<jupyter_start><jupyter_text>Method OutlineWe analyse the spontaneous emission (SE) rate of the EDTS layer in air and covered with water, denoted $1$ and $2$, respectively. Observing that that the SE rates calculated from PL measurements, $\Gamma^{1, 2} \equiv 1/\tau^{1,2}$, are composed of radiative, $\Gamma_{rad}$, and non-radiative, $\Gamma_{nrad}$, components gives$$\begin{align}\Gamma^1 &= \Gamma^1_{rad} + \Gamma^1_{nrad} \,,\\\Gamma^2 &= \Gamma^2_{rad} + \Gamma^2_{nrad} \approx \beta^2_1\Gamma^1_{rad} + \Gamma^1_{nrad}\end{align}$$where $\beta^2_1 = \Gamma^2_{rad}/\Gamma^1_{rad}$ is the theoretical radiative decay rate ratio and $\Gamma^1_{nrad} \approx \Gamma^2_{nrad}$ is assumed. The latter is justified as the non-radiative recombination processes in SiO$_2$ are due to short-range Forster energy transfer between the Er$^{3+}$ transition and the resonant hydroxyl groups, which are therefore not dependent on the external surrounding environment of the EDTS layer.Rewritten in matrix form this gives:$$\begin{bmatrix}    \Gamma^1 \\    \Gamma^2 \\\end{bmatrix}=\begin{bmatrix}            1 & 1\\    \beta^2_a & 1 \\\end{bmatrix}\begin{bmatrix}    \Gamma^1_{rad} \\    \Gamma^1_{nrad} \\\end{bmatrix} \,.$$The unknown radiative and non-radiative decay rates (in air) are therefore evaluated by solving$$\begin{bmatrix}    \Gamma^1_{rad} \\    \Gamma^1_{nrad} \\\end{bmatrix}=\begin{bmatrix}            1 & 1\\    \beta^2_a & 1 \\\end{bmatrix}^{-1}\begin{bmatrix}    \Gamma^1 \\    \Gamma^2 \\\end{bmatrix}$$The Quantum efficiency (q) is given by$$\begin{equation}q^x = \frac{\Gamma^x_{rad}}{\Gamma^x_{rad} + \Gamma_{nrad}} \left(= \frac{\tau^x_{mes}}{\tau^x_{rad}}\right) \,,\end{equation}$$and is therefore dependent on the medium ($x=a,w$) surrounding the EDTS layer.<jupyter_code>from photonics.misc import quantum_efficiency
# Waveguiding modes included
quantum_efficiency(12.200e-3, 12.082e-3, 0.9157)
# No Waveguiding
quantum_efficiency(tau1=12.200e-3, tau2=12.082e-3, beta21=1.623166626)<jupyter_output>Radiative decay rate    : 1.28 /s
Non-radiative decay rate: 80.68 /s 

Radiative lifetime      : 778.432 ms
Non-radiative lifetime  : 12.394 ms
Quantum Efficiency is 1.567%
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