# Method Outline
We 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.


```python
from photonics.misc import quantum_efficiency
```


```python
# Waveguiding modes included
quantum_efficiency(12.200e-3, 12.082e-3, 0.9157)
```

    Radiative decay rate    : -9.50 /s
    Non-radiative decay rate: 91.46 /s 
    
    Radiative lifetime      : -105.304 ms
    Non-radiative lifetime  : 10.933 ms
    Quantum Efficiency is -11.586%



```python
# No Waveguiding
quantum_efficiency(tau1=12.200e-3, tau2=12.082e-3, beta21=1.623166626)
```

    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%


f5c1fd9dee888aead90dc8ec5938c54b82eb6268
ipynb
Jupyter Notebook
examples/quantum efficiency.ipynb
mn14tm/lifetime
c851c96582b6f3c8863139a7656df98f95a350e9
RSA-MD
2017-06-15T15:26:02.000Z
2018-10-06T05:34:46.000Z
examples/quantum efficiency.ipynb
mn14tm/leeds-photonics
c851c96582b6f3c8863139a7656df98f95a350e9
RSA-MD
examples/quantum efficiency.ipynb
mn14tm/leeds-photonics
c851c96582b6f3c8863139a7656df98f95a350e9
RSA-MD
Qwen/Qwen-72B
1. YES
2. YES