# FDE solver analysis - Mode List and Deck

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 \( n_{\text{max(sub,clad)}} < n_{eff} < n_{wg} \).

** TE Fraction **

Two definitions for TE fractions are provided by 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, we only consider the fields parallel to the mode cross section). For propagation along the x/y direction (ie. "TE polarization fraction Ey/Ex"), this is similarly defined:

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

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

This definition is typically used in integrated optics, but for fibers, this also helps to determine the polarization of the mode. 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.

There are alternative definitions which are used more frequently for other applications/fields. For example, the "waveguide TE/TM fraction" is defined by the following equations:

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

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

where \( E_{\perp} \text{ and } H_{\perp} \) refers to the field component in the direction of propagation (ie. perpendicular to the mode cross section).

** 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.

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

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.

** Loss **

Loss is calculated based on the imaginary part of the effective index

$$ E(z) = E_{0} e^{-ink_{0}z} $$

$$ Loss(dB/m) = 10 \text{log}_{10} \bigg(\frac {P(z=1)} {P(z=0)} \bigg) = 10 \text{log}_{10} e^{-2 i n k_0 z} = \frac{10 n_{i} 4 \pi}{\lambda_{0} \text{log}_{10}} $$

where \$ n_i \$ is the imaginary part of the effective index.

### 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 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.