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.