# Standard optical permittivity material models in FDTD and MODE

This section describes the basic permittivity (or refractive index) material models supported by the Material Database. Model parameters can be edited in the Material property panel of the Material Database window.

## Sampled 3D data

The Sampled data model is used to import experimental material data. The experimental data can be imported from a text file with the Import data button. This method can be used to create a lossless material.

There are two types of sampled data models available: *Sampled 3D data* and *Sampled 2D data*. Sampled 2D data can be used for importing the surface conductivity data from 2D materials such as graphene. For more information about the Sampled 2D data material, see Material conductivity models.

Note: The Sampled data material definition uses an automatic fitting routine to generate a multi-coefficient material model of the experimental data over the frequency range specified by the source. The fits can be checked and adjusted in the Material Explorer. |

- TOLERANCE: The desired RMS error between the permittivity of the experimental data and the material fit. The fitting routine will use the least number of coefficients that produce a fit with an RMS error less than the tolerance.
- MAX COEFFICIENTS: The maximum number of coefficients allowed to be used in the material fit. More coefficients can produce a more accurate fit, but will make the simulation slower.

The following advanced options can be set in the Material Explorer:

- MAKE FIT PASSIVE: Set to be true to prevent the fit from having gain at any frequency. By default this is true in order to prevent diverging simulations.
- IMPROVE STABILITY: If this setting is true, the fitting routine restricts the range of coefficients in the fit in order to reduce numerical instabilities which cause simulations to diverge.
- IMAGINARY WEIGHT: Increasing the weight increases the importance of the imaginary part of the permittivity when calculating a fit. A weight of 1 gives equal weight to the imaginary and real parts of the permittivity.
- SPECIFY FIT RANGE: Set to true to decouple the bandwidth used to generate the material fit and the source bandwidth. This option is used in parameter sweeps where the source frequency is changed, and where it is important to keep the material parameters constant over the whole parameter sweep. The fit range should cover the simulation bandwidth.
- BANDWIDTH RANGE: Bandwidth to be used for the fit when Specify Fit Range is true.

Note: Sampled data can still be used, for backward compatibility; however, it is recommended to use Sampled 3D data whenever possible. |

### Examples and more information

Creating sampled data materials, Checking material fits with the material explorer

## Dielectric

The Dielectric model is used to create a material with a constant real index. This material will have the specified index at all frequencies (non-dispersive).

- REFRACTIVE INDEX: The refractive index of the material. Must be >= 1.

## (n,k) material

The (n,k) material model is used to create a material with a specific value of n and k at a single frequency.

- REFRACTIVE INDEX: Real part of the index at the center frequency of the simulation. Must be > 0.
- IMAGINARY REFRACTIVE INDEX: Imaginary part of the index at the center frequency of the simulation. Positive values correspond to loss, negative values will produce gain.

### Examples and more information

n,k material model, Checking material fits with the material explorer

NOTE: Single frequency simulations only! This type of material model should only be used for single frequency simulations. The implementation of the (n,k) material model is such that the material properties will only be correct at the center frequency of the simulation. |

## Conductive 3D

The Conductive model is used to create a material with the the following relative permittivity.

$$ \varepsilon_{\text{total}}(f) = \varepsilon + i \frac{\sigma}{2\pi \cdot f\varepsilon_{0}}$$

- \(\varepsilon\): permittivity
- \(\sigma\): conductivity in units of (Ωm)-1

Note: Comparison with PEC As the conductivity becomes very large, the performance of this model approaches the ideal PEC (Perfect Electrical Conductor) model described below. |

## Plasma (Drude)

The Plasma model is used to create a material with the the following relative permittivity.

$$ \varepsilon_{\text{total}}(f) = \varepsilon - \frac{\omega^2_p}{2\pi \cdot f(i\nu_c + 2\pi \cdot f)}$$

- \(\varepsilon\): permittivity
- \(\omega_p\): plasma resonance in units of rad/s
- \(\nu_c\): plasma collision in units of rad/s

## Debye

The Debye model is used to create a material with the the following relative permittivity.

$$ \varepsilon_{\text{total}}(f) = \varepsilon - \frac{\varepsilon_{\text{debye}}\cdot \nu_c}{\nu_c + i2\pi \cdot f}$$

- \(\varepsilon\): permittivity

- \(\varepsilon_{Debye}\): Debye permittivity
- \(\nu_c\): Debye collision in units of rad/s

## Lorentz

The Lorentz model is used to create a material with the the following relative permittivity.

$$ \varepsilon_{\text{total}}(f) = \varepsilon - \frac{\varepsilon_{\text{lorentz}}\cdot \omega^2_0}{\omega^2_0 -2i\delta_0 2\pi \cdot f + (2\pi \cdot f)^2}$$

- \(\varepsilon\): permittivity
- \(\varepsilon_{lorentz}\): Lorentz permittivity
- \(\omega_0\): Lorentz resonance in units of rad/s
- \(\delta_0\): Lorentz linewidth in units of rad/s

NOTE: Lorentz model reference Kurt Oughstun and Natalie Cartwright, "On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion," Opt. Express 11, 1541-1546 (2003) |

## Sellmeier

The Sellmeier model is used to create a material defined by the following formula. The C coefficients have dimensions of micrometers squared (mm2).

$$ \varepsilon_{\text{total}(\lambda)} = A_1 + \frac{B_1\lambda^2}{\lambda^2-C_1} + \frac{B_2\lambda^2}{\lambda^2-C_2} + \frac{B_3\lambda^2}{\lambda^2-C_3} $$

NOTE: Single frequency simulations only! This type of material model should only be used for single frequency simulations. The implementation of the Sellmeier model is such that the material properties will only be correct at the center frequency of the simulation. |

## PEC

A Perfect Electrical Conductor (PEC). The Electric field within this material must be zero. It will have 100% reflection and 0% absorption. There are no parameters for this model.

### Understanding the refractive index of PEC as reported in the Material Explorer and Refractive index monitors

The refractive index of PEC is not well defined. This can be understood by considering the PEC material as a conductive material with an infinite conductivity. As the conductivity sigma goes to infinity, the permittivity goes to infinity. Having the Material explorer and Refractive index monitor return infinite values is not ideal, so they report the permittivity as eps = 1+ 1e6i, meaning the refractive index is reported as sqrt(1+ 1e6i)=707+707i. It is important to understand that these values are only for the purpose of reporting by the Material Explorer and Refractive index monitors. The actual solver engine uses an ideal model (ie. infinite conductivity).

$$ \varepsilon_{\text{total}}(f) = \varepsilon + i \frac{\sigma}{2\pi \cdot f\varepsilon_{0}},\sigma \rightarrow \infty$$

$$ \varepsilon_{\text{total}}(f) = \varepsilon + i \infty$$

Note: Spatial absorption measurements It is possible for the difference between the permittivity used in the solver (infinite) and the permittivity reported by the Refractive index monitor (1e6) to cause problems with the spatial absorption monitors. This will not be an issue when measuring the total absorbed power with a box of monitors. Contact Lumerical support for further details. |

## Analytic material

The analytic material model allows the user to enter an equation for the real and imaginary part of the permittivity or refractive index which can depend on the predefined variables listed below. For an example, see the page.

### Examples and more information

Simple analytic material model, Graphene material (volumetric approach), Checking material fits with the material explorer

The predefined variables that can be used in the equations for "real" and "imaginary" are:

- f: the frequency in the specified frequency units
- l0: the free space wavelength in the specified length units
- w: 2*pi*f in the specified units
- k0: 2*pi/l0 where l0 is in the specified length units
- pi: the number pi
- c: the speed of light in a vacuum, ALWAYS in SI units, ie, always equal to 3e8
- x1,...,x10: numeric values that represent a parameter of interest

Additional resources: Advanced material models, Flexible material plugin framework