# Lempel-Ziv Complexity of Photonic Quasicrystals

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Fibonacci Quasicrystals

## 3. Spectral Measures of Generalized Fibonacci Quasicrystals

## 4. Lempel-Ziv Complexity

## 5. Complexity of Generalized Fibonacci Quasicrystals

_{3}AlF

_{6}) and zinc selenide (ZnSe), with refractive indices ${n}_{L}=1.34$ and ${n}_{H}=2.568$, respectively, at $\lambda =0.65\phantom{\rule{4pt}{0ex}}\mathsf{\mu}\mathrm{m}$. This is a standard choice and, although it is a simple example, it allows one to work out easily the details of the method, which can be immediately extended to other media and spectral regions.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Illustrations of the different quasiperiodic chains considered in this work. In the top panel, we have the periodic case (40 letters). In the mid panel, the Olympic-metal family Fibonacci sequence (FS) (h,1), with (from left to right) $h=1$ (golden mean, 34 letters), $h=2$ (silver mean, 41 letters), and $h=3$ (bronze mean, 43 letters). In the bottom panel, the non-Olympic-metal family FS (1, ℓ), with $\ell =2$ (copper mean, 43 letters) and $\ell =3$ (nickel mean, 40 letters).

**Figure 2.**Normalized power spectrum for words up to 1500 letters for different generalized Fibonacci sequences. In the left panel, for the Olympic-metal family FS(h,1), with $h=1,2,$ and 3. In the right panel, for the non-Olympic-metal sequences FS(1,2) and FS(1,3).

**Figure 3.**Behavior of the transmittance as a function of the LZ complexity for the periodic system and the generalized Fibonacci families indicated in the inset. We consider generations up to 1600 letters.

**Figure 4.**(

**Left**) Lempel-Ziv (LZ) complexity versus the number of letters in the words of several generations for the periodic systems and the generalized Fibonacci quasicrystals indicated in the insets; (

**Right**) Transmittance for the same cases as before, as a function of the number of letters ${w}_{\alpha}$.

**Table 1.**Thicknesses yielding minimum average transmittance over all the generations in the generalized Fibonacci quasicrystals indicated in the first column.

System | Metal | ${\mathit{n}}_{\mathit{L}}{\mathit{d}}_{\mathit{L}}/\mathit{\lambda}$ | ${\mathit{n}}_{\mathit{H}}{\mathit{d}}_{\mathit{H}}/\mathit{\lambda}$ |
---|---|---|---|

Periodic | - | 0.2500 | 0.2500 |

FS(1,1) | Gold | 0.2251 | 0.3417 |

FS(2,1) | Silver | 0.2311 | 0.3809 |

FS(3,1) | Bronze | 0.3416 | 0.2459 |

FS(1,2) | Copper | 0.1249 | 0.2501 |

FS(1,3) | Nickel | 0.2451 | 0.2277 |

System | Metal | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{b}}_{0}$ | ${\mathit{b}}_{1}$ | ${\mathit{c}}_{0}$ | ${\mathit{c}}_{1}$ |
---|---|---|---|---|---|---|---|

Periodic | - | −17.917 | 17.301 | 0.54422 | 0.04718 | −1.82860 | 0.66697 |

FS(1,1) | Gold | −29.779 | 52.116 | 0.46257 | 0.00998 | −0.35174 | 0.41042 |

FS(2,1) | Silver | −22.297 | 22.590 | 0.71580 | 0.01821 | 0.29576 | 0.27829 |

FS(3,1) | Bronze | −10.953 | 9.847 | 0.90351 | 0.02264 | 0.28379 | 0.16478 |

FS(1,2) | Copper | −23.922 | 41.833 | 0.44177 | 0.00980 | −1.27180 | 0.32427 |

FS(1,3) | Nickel | −18.687 | 24.282 | 0.57597 | 0.00876 | 0.16579 | 0.14514 |

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**MDPI and ACS Style**

Monzón, J.J.; Felipe, A.; Sánchez-Soto, L.L. Lempel-Ziv Complexity of Photonic Quasicrystals. *Crystals* **2017**, *7*, 183.
https://doi.org/10.3390/cryst7070183

**AMA Style**

Monzón JJ, Felipe A, Sánchez-Soto LL. Lempel-Ziv Complexity of Photonic Quasicrystals. *Crystals*. 2017; 7(7):183.
https://doi.org/10.3390/cryst7070183

**Chicago/Turabian Style**

Monzón, Juan J., Angel Felipe, and Luis L. Sánchez-Soto. 2017. "Lempel-Ziv Complexity of Photonic Quasicrystals" *Crystals* 7, no. 7: 183.
https://doi.org/10.3390/cryst7070183