This page provides more information on the Mode List and the Deck part of the Eigensolver analysis window.

## Mode list

The mode list shows all of the modes that were calculated in the MODAL ANALYSIS tab along with their effective index, loss (if applicable), effective area and TE fractions.

### Effective Index

The effective index is through the velocity phase relation.

$$ n_{eff} = \frac{c}{v} = \frac{\beta}{k_0} $$

Similar to the bulk index definition, except that geometric contributions are taken into account. Since the mode must be guided the effective index is bounded by the indices such that \( n_{\text{max(sub,clad)}} < n_{eff} < n_{wg} \).

### Loss (dB/m)

An electric field propagating in the positive \(z\)-direction in a medium with a complex refractive index of \(n+i\kappa\) can be expressed as follows:

$$E(z) = e^{i2\pi(n+i\kappa)z/\lambda_0}$$

[[NOTE:]] The \(\kappa\) in the above expression refers to the imaginary part of the complex refractive index, not the wavevector, \(k = 2\pi(n+i\kappa)/\lambda_0\). |

The propagation loss in dB/m for a mode propagating in the \(z\)-direction is defined as

$$loss = -10 \log_{10}(\frac{P(z)|_{z = \ 1\ m}}{P(z)|_{z = \ 0\ m}}) = -10 \log_{10}(\frac{|E(1)|^2}{|E(0)|^2})=-20 \log_{10}(\frac{|E(1)|}{|E(0)|})$$

Combining the formulas gives

$$loss = -20 \log_{10}(e^{-2\pi\kappa/\lambda_0})$$

### TE Polarization Fraction

Two definitions for polarization fractions are provided by MODE. The TE Polarization Fraction definition is typically used in integrated optics, but for fibers, this also helps to determine the polarization of the mode.

The "TE polarization fraction Ex" for propagation along the z-direction is defined by the following equation:

$$\text{TE polarization fraction (Ex)} = \frac{\int |Ex|^2 dxdy}{\int (|Ex|^2+|Ey|^2)dxdy} $$

where \( |Ex|^2+|Ey|^2 \) corresponds to \( |E_{||}|^2 \) (since we are considering the polarization of the modes, here we only consider the fields parallel to the mode cross section). In this case, if TE polarization fraction (Ex) = 100%, this mode is considered pure TE-polarized. In contrast, 0% refers to a pure TM-polarized mode. Note that some modes may not be perfectly polarized in one direction and in those cases you may find quasi-polarized modes in the Mode List.

For propagation along the x direction, ie., "TE polarization fraction Ey":

$$ \text{TE polarization fraction (Ey)} = \frac{\int |Ey|^2 dydz}{\int (|Ey|^2+|Ez|^2) dydz} $$

For propagation along the y direction, ie., "TE polarization fraction Ex":

$$ \text{TE polarization fraction (Ex)} =\frac{\int |Ex|^2 dxdz}{\int (|Ex|^2+|Ez|^2) dxdz} $$

Note that this definition is arbitrary (since we allow for propagation in any direction), the user may have to look carefully at the field components if unusual orientations of the Eigenmode Solver are used.

### Waveguide TE/TM Fraction

There are alternative definitions that are used more frequently for other applications. The waveguide TE/TM fraction indicates the amount of E/H field in the direction of propagation. It is equal to the integrated transverse field intensity divided by the integrated total field intensity. A mode with a waveguide TE/TM fraction of 100%/100% is a TEM mode.

The "waveguide TE/TM fraction" is defined by the following equations:

$$\text{waveguide TE fraction } = 1-\frac{\int\left|E_{\perp}\right|^{2} dA_{||}}{\int\left(|E|^{2}\right) dA_{||}} $$

$$ \text{waveguide TM fraction } = 1-\frac{\int\left|H_{\perp}\right|^{2} dA_{||}}{\int\left(|H|^{2}\right) dA_{||}} $$

where \( E_{\perp} \) and \( H_{\perp} \) refer to the field components perpendicular to the mode cross section (ie. in the direction of propagation), \(A_{||}\) refers to the integration area on the mode cross section plane.

### Effective Area

Effective area is a quantitative measure of the modal area. As with TE fraction, there are a number of possible similar definitions which can be easily misinterpreted. We have adopted the definition developed by G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007). This method has the benefit of not making any assumptions about the shape of the field distribution beforehand.

$$\text{Effective Area} = \frac{(\int_{}^{}|E|^{2}dA)^{2} }{\int_{}^{}|E|^{4}dA}$$

The effective area defined here is similar to the Landau, and Lishitz method, based on the ratio of a mode's total energy density per unit length and its peak energy density.

For Gaussian beams, and fiber optic modes the most cited quantity is the beam waist w(z) or equivalently mode field diameter (MFD). The MFD corresponds to twice the radial distance, where \( I(r) = 1-e^{-2}\approx 13.5\)%. In with the comparison normal distribution this radial value would correspond to \( f( x \approx 3 \sigma )\), and is related to the full width half maximum (FWHM) as \( w(z) \approx 1.18 \text{ FWHM} \). The effective area method cited above is typically in the range of 95-104% of what would be found using through the circle described by the MFD.

## Object tree

The modes in the mode list described above, as well as results from Frequency analysis, are also stored in the "data" analysis group under the Eigenmode Solver simulation region. One can study these results using the Lumerical Visualizer.

## Deck

The deck section of the analysis window shows all the stored D-CARD data.

D-CARDs are data repositories that store all the information necessary to perform complex operations such as modal overlaps. D-CARDs on the deck can be modes calculated in MODE and copied from the mode table (by right-clicking on the desired mode) or loaded from a file. Also, D-CARDs can be imported from ASAPTM output, they can be imported from FDTD, or they can be a default Gaussian beam. Each D-CARD contains a large amount of data, such as vectorial electromagnetic fields, frequency vectors, and effective indices. By manipulating this information as a D-CARD, the user can easily view, store, manage and analyze large amounts of data with a few mouse clicks.

NOTE: Calculated analysis data is not lost if you close the analysis window The analysis window can be closed by pressing the close button (red cross at the top right side of the window in the image above). When you re-open the window (for example by pressing the Run Active Simulation button ) all the data that was calculated before the window was closed will be available. Also, it is possible to open CAD and examine the properties of the structures and the simulation region when analyzing a particular mode. However, to make changes to these structures MODE must first be switched back into Layout Mode. This can be done by pressing the Switch to Layout button in CAD. When you switch to layout all local data that has been calculated for the current structure will be lost. This ensures that if you have a file with data stored in the analysis tab, then that data will belong to the structure which is in the CAD environment. |