## Full parameter study of multilayer coating for thermal applications

We present the formulation of the reversal engineering problem for designing coherent thermal radiators with a planar multilayer stacks coating.

In the context of this work, control of thermal coherence radiation means that we can design a surface finishes exhibiting a predefined spectral directional emittance (or emissivity) and reflectance (or reflectivity): `e(\lambda,\theta)` and `r(\lambda,\theta)` . The reverse engineering problem is defined by two main components: i) the objective function and ii) the optimisation solver.

### Objective function

The objective function provides a single numerical value by comparing the optical properties of a given planar multilayer structure with the desired optical properties. This number is an objective weight representing the quality of the solution. Mathematically such a function can be written as:

`r_{"T"}=1-e_{"T"}`.

The Figure 1 contains a graphical representation of the target emittance together with the auxiliary parameters introduced above to describe it. In addition, the Figure contains a polar plot for three discrete values of `lambda`. The lobes results from the spatial coherence of the emittance while a small `\Delta\lambda` results from the temporal coherence. In summary, the target emittance/reflectance is defined by means of a set of nine parameters.

### Optimisation solver

The optimisation problem is solved using a generic Genetic Algorithm consisting of only three genetic operators: reproduction, cross-over and mutation. Although many implementations of this problem are possible, we have adopted a pure integer one.

The thermal radiator is described by a given number of layers with each of the layers made of a given material and presenting a thickness. A straight way of implementing this problem consists in using an integer representation to define the materials of each layer and a real representation for the layer thickness. In this work, we describe also the layer thickness with integers. We define a minimum layer thickness unit and let the optimiser algorithm to determine the number of units building each layer. This approach requires to use a thin layer thickness units and consider a large number of layers. More details can be found in [1] and [2].

### Results

We are updating regularly the results obtained from the optimisation. Please visit the results web page and check what are the limits of this technology for controlling the coherent thermal radiation.

- [1]
Ben-Abdallah, P., Microstructured Radiators, European Space Agency, the Advanced Concepts Team,
Ariadna Final Report
**06-9501a**, (2007). - [2]
Drevillon, J. and Ben-Abdallah, P.,
*Ab initio*design of coherent thermal sources, Journal of Applied Physics**102**, 114305 (2007).

**References:**