# In-Fiber All-Optical Fractional Differentiator Using an Asymmetrical Moiré Fiber Grating

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## Abstract

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## 1. Introduction

_{2}laser irradiation [15], and arc-discharge [16], where the phase shift is due to a change in the effective index induced in the local region of the FBG. Recently, the infrared femtosecond laser-writing method has been proposed to fabricate PS-FBGs [17]. However, the ultimate requirement for extremely high precision for the manufacturing setups and excellent quality for the employed laser beam would restrain this approach from being applied in industry for mass output. The reader can find more details on how to insert a phase shift into a FBG in [4], and the references therein.

## 2. Theoretical Concepts

## 3. Results and Discussion

_{0}, Figure 8 shows the natural logarithm of the deviation factor ($\mathrm{ln}{D}_{0.51}$). For this purpose, several input signals with time widths in the range of 5 ps to 155 ps and the same 0.51th-order differentiator as before were used. In this last figure, it can be seen that the minimum occurs when the time width is close to 40 ps, and that the deviation factor is not significantly increased by a small input bandwidth detuning. It should also be considered that the transmission drop is not exactly zero (see Figure 6a). Therefore, low-frequency components are not sufficiently rejected. In addition, the phase shift occurred over a bandwidth of 0.8 GHz (see Figure 6b), which represents 9% as compared with the FWHM (full-width at half maximum) of the Fourier transform of the pulse under test. It is important to mention that the length of the grating $L$ is related to the bandwidth. Generally, shorter devices (i.e., with shorter lengths) can process higher bandwidths [7].

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Plot of the operator ${H}_{n}\left(\nu \right)$ for the 0.33th-order differentiator frequency response in amplitude, in baseband frequency.

**Figure 2.**(

**a**) Uniform fiber Bragg gratings with slightly different periods ${\mathsf{\Lambda}}_{1}$ and ${\mathsf{\Lambda}}_{2}$; (

**b**) Schematic of the Moiré grating structure with rapidly varying ${\mathsf{\Lambda}}_{s}$ and slowly varying ${\mathsf{\Lambda}}_{c}$ envelopes.

**Figure 3.**Refractive index modulation of AMFG generated by superimposing two gratings with slightly different periods, ${\mathsf{\Lambda}}_{1}$ and ${\mathsf{\Lambda}}_{2}$.

**Figure 5.**Fractional differentiation order n as a function of the relative grating length $\raisebox{1ex}{${L}_{1}$}\!\left/ \!\raisebox{-1ex}{$L$}\right.$.

**Figure 6.**(Color online) Ideal (blue-dashed curve) and proposed (black-solid curve) 0.51th-order differentiator frequency response in (

**a**) amplitude and (

**b**) reflection phase, in baseband frequency. Amplitude (red-dotted curve) of the pulse under test is also shown in both figures normalized to unity.

**Figure 7.**Simulated (solid curve) and ideal (dashed curve) time response of the designed 0.51th-order differentiator. The input signal is also displayed (dotted curve); all signals were normalized to unity.

**Figure 8.**Natural logarithm of the deviation factor D between the theoretical and proposed 0.51th-order temporal differentiator for different temporal widths ${T}_{0}$.

**Figure 9.**Ideal 0.51th-order differentiator (dashed curve) and simulated 0.55th-order differentiator (solid curve) time response. The input signal is also displayed (dotted curve); all signals were normalized to unity.

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**MDPI and ACS Style**

Tendela, L.P.; Cuadrado-Laborde, C.A.; Andrés, M.V. In-Fiber All-Optical Fractional Differentiator Using an Asymmetrical Moiré Fiber Grating. *Fractal Fract.* **2023**, *7*, 291.
https://doi.org/10.3390/fractalfract7040291

**AMA Style**

Tendela LP, Cuadrado-Laborde CA, Andrés MV. In-Fiber All-Optical Fractional Differentiator Using an Asymmetrical Moiré Fiber Grating. *Fractal and Fractional*. 2023; 7(4):291.
https://doi.org/10.3390/fractalfract7040291

**Chicago/Turabian Style**

Tendela, Lucas P., Christian A. Cuadrado-Laborde, and Miguel V. Andrés. 2023. "In-Fiber All-Optical Fractional Differentiator Using an Asymmetrical Moiré Fiber Grating" *Fractal and Fractional* 7, no. 4: 291.
https://doi.org/10.3390/fractalfract7040291