Quantities of optical modes#
TE/TM Polarization Fraction#
\[ \begin{align}\begin{aligned}
\mathrm{TEfrac}
&=
\frac{
\int \left| E_{x_1} \right|^2 \mathrm{d}x\mathrm{d}y
}{
\int \left| E_{x_1} \right|^2 + \left| E_{x_2} \right|^2 \mathrm{d}x \mathrm{d}y
}\\ \mathrm{TMfrac}
&=
\frac{
\int \left| E_{x_2} \right|^2 \mathrm{d}x\mathrm{d}y
}{
\int \left| E_{x_1} \right|^2 + \left| E_{x_2} \right|^2 \mathrm{d}x \mathrm{d}y
}
\end{aligned}\end{align} \]
Loss per meter [dB/m]#
\[\begin{split}
\text{Loss at }x_3\text{ [dB]}
&=-10 \log_{10} \frac{\left|E(x_3)\right|^2}{\left|E(x_3=0)\right|^2}
\\
&=-20 \log_{10} \frac{\left|E(x_3)\right|}{\left|E(x_3=0)\right|}
\\
&=-20 \log_{10} \mathrm{e}^{\Im\beta x_3}
\\
&=-20 \frac{\log_{\mathrm{e}} \mathrm{e}^{\Im\beta x_3}}{\ln 10}
\\
&=\frac{-20}{\ln 10} \Im\beta x_3
\\
\\
\text{Loss [dB/m]}
&=
\frac{-20}{\ln 10} \Im\beta \, 1\mathrm{m}
\end{split}\]
Effective Area#
As defined in [1]
\[
A_{\text{eff}}
=
\frac{
\left( \int \left| \vec{\mathcal{E}} \right|^2 \mathrm{d}A \right)^2
}{
\int \left| \vec{\mathcal{E}} \right|^4 \mathrm{d}A
}
\]
Confinement coefficient#
As defined in [2] (and generalized for varying refractive indices in the active area)
\[
\Gamma
=
\frac{
c \epsilon_0 \int n(\vec{x}) \left| \vec{\mathcal{E}} \right|^2 \mathrm{d}A
}{
\left( \int \vec{\mathcal{E}}^* \times \vec{\mathcal{H}}
+
\vec{\mathcal{E}} \times \vec{\mathcal{H}}^*
\mathrm{d}A \right) / 2
}
\]
Overlap coefficient#
\[
c_{\nu\mu}
=
\frac{
\int \vec{\mathcal{E}}_\nu^* \times \vec{\mathcal{H}}_\mu
+
\vec{\mathcal{E}}_\nu \times \vec{\mathcal{H}}_\mu^* \mathrm{d}A
}{
\prod_{i=\{\mu,\nu\}}
\sqrt{
\int \vec{\mathcal{E}}_i^* \times \vec{\mathcal{H}}_i
+
\vec{\mathcal{E}}_i \times \vec{\mathcal{H}}_i^* \mathrm{d}A
}
}
=
c_{\mu\nu}^*
\]
Characteristic impedance#
https://ieeexplore.ieee.org/document/108320
Power and current:
\[ \begin{align}\begin{aligned}
P_k
=
\delta_{jk}
\int
\left(
\vec{\mathcal{E}}_j^* \times \vec{\mathcal{H}}_k
\right) \cdot \hat{x}_3\\ I_{zik} = \oint_{C_i} \mathcal{H} \ cdot
\end{aligned}\end{align} \]
Characteristic impedance:
\[ \begin{align}\begin{aligned}
P = I^T Z_c I\\ Z_c = [I^{-1}]^T P I^{-1}
\end{aligned}\end{align} \]
[1]
Govind Agrawal. Nonlinear Fiber Optics. Academic Press, Amsterdam, Boston, edition, 2019. ISBN 978-0-128-17043-4.
[2]
Jacob T. Robinson, Kyle Preston, Oskar Painter, and Michal Lipson. First-principle derivation of gain in high-index-contrast waveguides. Optics Express, 16(21):16659, October 2008. URL: https://doi.org/10.1364/oe.16.016659, doi:10.1364/oe.16.016659.