# Bragg Mirrors for Thermal Waves

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## Abstract

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## 1. Introduction

## 2. Thermal Waves—Cattaneo-Vernotte Equation

## 3. Thermal Superlattice

## 4. Thermal Bragg Mirrors

#### Tuning of the Thermal Reflectance

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Finite layered system made of consecutive layers with different thermal properties. Each layer is characterized by its thickness d, thermal speed $\upsilon $ and the thermal diffusivity $\alpha $. The system is bounded by a substrate and outer medium. (

**b**) Semi-infinite system. The substrate is removed and the layered system extends indefinitely.

**Figure 2.**Reflectance R between the medium labeled material 1 and a half space made of Al (

**a**), material 2 (

**b**). The properties of the different materials are given in Table 1. As explained in the text, the reflectance for thermal waves is $R=\left|r\right|$ (solid) rather than $R={\left|r\right|}^{2}$ (dashed) as in other wave phenomena. For metals that have a very small relaxation time, they behave as perfect reflectors.

**Figure 3.**Penetration length $\delta $ (in mm) as a function of frequency for the materials shown in Table 1. The dotted lines indicate the asymptotic value given by $\delta =2\alpha /v$.

**Figure 4.**For a system with unit cells made of alternating layers of material 1 and material 2 (see Table 1) and both of thickness $d=100$ $\mathsf{\mu}$m, we present in (

**a**–

**c**) the reflectance $R=\left|r\right|$ as a function of frequency for different numbers of unit cells n in a solid line; the dashed line represents the semi-infinite periodic case. In (

**d**), the heat flux into the system at $x=0$ is shown as function of frequency and n. The flux is normalized to the incident flux ${q}_{0}$. The stop-band is seen even for a few number of unit cells. The dotted lines indicate the position of the first stop-band for the semi-infinite crystal.

**Figure 5.**Normal heat flux in a contour plot for a system with unit cells made of alternating layers of material 1 and material 2 assuming that they have relaxation time equal to zero, which corresponds to the Fourier law case.

**Figure 6.**(

**a**–

**c**) Reflectance and normalized (

**d**) heat flux of a system with unit cell m1-m2-Al-m2 (1-2-Al-2). Layers of Material 1 and 2 have a thickness of $d=100\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m while for that of Al ${d}_{al}=1\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m.

Material | Thermal Conductivity $\mathit{\kappa}$ (Wm${}^{-1}$ K${}^{-1}$) | Specific Heat Capacity ${\mathit{c}}_{\mathit{V}}$ (J kg${}^{-1}$ K${}^{-1}$) | Mass Density $\mathit{\rho}$ (Kg m${}^{-3}$) | Response Time $\mathit{\tau}$ (s) |
---|---|---|---|---|

epidermis (m1) | 0.235 | 3600 | 1500 | 1 |

dermis (m2) | 0.445 | 3300 | 1116 | 20 |

Al | 237 | 921 | 2707 | $2\times {10}^{-10}$ |

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**MDPI and ACS Style**

Camacho de la Rosa, A.; Becerril, D.; Gómez-Farfán, M.G.; Esquivel-Sirvent, R. Bragg Mirrors for Thermal Waves. *Energies* **2021**, *14*, 7452.
https://doi.org/10.3390/en14227452

**AMA Style**

Camacho de la Rosa A, Becerril D, Gómez-Farfán MG, Esquivel-Sirvent R. Bragg Mirrors for Thermal Waves. *Energies*. 2021; 14(22):7452.
https://doi.org/10.3390/en14227452

**Chicago/Turabian Style**

Camacho de la Rosa, Angela, David Becerril, María Guadalupe Gómez-Farfán, and Raúl Esquivel-Sirvent. 2021. "Bragg Mirrors for Thermal Waves" *Energies* 14, no. 22: 7452.
https://doi.org/10.3390/en14227452