# Temperature dependent refractive index models

This section describes the models used to calculate perturbation in refractive index for optical simulations in MODE and FDTD based on imported temperature profile data from HEAT simulations.

## Linear sensitivity model

The simpler model used to calculate index perturbation due to temperature variation assumes a linear dependency. The complex refractive index of a material at temperature T is given by,

$$ n + ik = (n_{ref} + \Delta n) + i(k_{ref} + \Delta k) $$

where, n is the real and k is the imaginary part of the refractive index at temperature T, nref is the real and kref is the imaginary part of the unperturbed refractive index at temperature Tref, and Δn is the change in the real part and Δk is the change in the imaginary part of the refractive index. The values of Δn and Δk are given by,

$$ \Delta n = \frac{dn}{dT} (T - T_{ref}) $$

$$ \Delta k = \frac{dk}{dT} (T - T_{ref}) $$

where, dn/dT and dk/dT are the rate of change in the real and imaginary part of refractive index, respectively. In the linear sensitivity model, the rate of change in refractive index is assumed to be constant at reference temperature Tref and is used as a material property in the "index perturbation" type material.

## Nonlinear (tabular) model

The second model available for modeling index perturbation due to temperature variation allows for the modeling of nonlinear sensitivity. The nonlinear variation of refractive index can be given as an input in tabular form to the "index perturbation" type material. The effective index of the material at temperature T is then given by,

$$ n + ik = (n_{ref} + \Delta n) + i(k_{ref} + \Delta k) $$

where, n is the real and k is the imaginary part of the refractive index at temperature T, nref is the real and kref is the imaginary part of the unperturbed refractive index at temperature Tref (300 K for the materials in the default material database), and Δn is the change in the real part and Δk is the change in the imaginary part of the refractive index. For a nonlinear temperature sensitivity, Δn and Δk are functions of temperature T.

NOTE: At T = Tref, the values of Δn and Δk should be zero. |