# Spectral Characteristics of Square-Wave-Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{2}laser [2,17], femtosecond laser [18,19], and so on. Exerting periodical mechanical stress on a few-mode fiber could induce coupling between two copropagating core-guided modes, but such LPFGs could persist when mechanical stress was released [20]. A periodic taped structure induced by an electrical arc or structuring a fiber with a CO

_{2}laser could also realize the fabrication of LPFGs [21,22,23,24], but these LPFGs were relatively fragile because a few sections of the fiber became thinner just like microfibers [25,26,27]. UV lasers could inscribe relatively stable LPFGs by the direct-writing method or with the help of amplitude masks, but sufficient photosensitivity of fibers is necessary, so fibers should usually be hydrogen loaded in advance [28]. A femtosecond laser is another effective tool to fabricate LPFGs [29,30,31] with which the fiber is intact after fabrication, and the photosensitivity of fibers is not necessary.

^{–4}) and took up the majority of the core area [32,33]. Type I LPFGs might degenerate in high-temperature environments. Another kind of LPFG inscribed by a femtosecond laser, namely type II LPFGs, is completely different. The RI profile is negative (magnitude: 1 × 10

^{–3}) and usually highly localized. Even when the temperature is more than 1000 °C, type II gratings can still keep their characteristics [34]. Their fine temperature tolerance makes LPFGs good candidates in high temperature sensing and high-power fiber laser systems. Square-wave-modulated type II LPFGs are easy to fabricate, but the insertion loss is relatively higher than that of type I, and the spectrum is in chaos. Up to now, few studies have interpreted their characteristics.

## 2. Theory

_{1}and β

_{2}are the propagating constant of the two coupling modes, and Λ is the grating period of the LPFG. According to the relation of β = 2πn

_{eff}/λ, the resonant wavelength can be given by

_{eff}

_{,1}and n

_{eff}

_{,2}are the effective RI of the two different modes. For single-mode operation, the core-guided fundamental mode can only be coupled to cladding modes, and a bunch of resonant wavelengths appear in the transmission spectrum. Considering the symmetry, the fundamental mode mainly couples to axisymmetric cladding modes LP

_{0n}. Figure 1 shows the phase-matching condition of the first-order resonance (m = 1) when the grating period is 560 μm. The horizontal line represents 2π/Λ, and the intersection points define the resonant wavelengths. Simulation results indicate that LP

_{01}mode couples to cladding mode LP

_{02}at the wavelength of 1435 nm, to LP

_{03}at 1475 nm, and LP

_{04}at 1560 nm.

_{n}is the duty cycle of the nth square wave pulse (n = 1, 2, …, 72). The Fourier transform of Equation (4) can be written as:

## 3. Experiment

_{04}) saturates prior to the lower-order cladding mode (such as LP

_{03}and LP

_{02}). Because the LPFG does not strictly localize in the center of the core, the cross-RI profile takes on an axisymmetric rather than a circularly symmetric form; thus, birefringence becomes obvious when the LPFG is long. In the LPFG fabrication process, mode coupling is a dynamic process, for which energy in cladding modes may convert back to the fundamental mode when the grating reaches a certain length (this process is named overcoupling). As shown in Figure 6, after the overcoupling process, the resonance re-enhances, and birefringence occurs in the meantime.

_{02}, LP

_{03}, and LP

_{04}cladding mode at around 1425, 1480, and 1550 nm, respectively, which is close to the theoretical analysis in Figure 1, and the error mainly comes from the neglect of material dispersion, the higher-order harmonic resonance of the grating is also visible. The occurrence of higher-order harmonic resonance creates chaos over the transmission spectrum of the LPFG. What has to be pointed out is that the higher-order resonance generated by the LPFG with a duty cycle of 50% is weaker than the other two LPFGs, which can be explained by what we discussed in Figure 2 that the harmonic number of square waves with a duty cycle of 50% is less than the other duty cycle.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Schematic of a square wave. (

**b**) Spatial spectrum characteristics of a square wave with different duty cycles (color bar indicates the amplitude of each harmonic).

**Figure 4.**(

**a**) Waveform of a square wave with a linearly growing duty cycle. (

**b**) Spatial spectrum characteristics of a square wave with a linearly growing duty cycle.

**Figure 5.**(

**a**) Schematic of LPFG fabrication. (

**b**) RI modulation generated point-by-point (magnification: 100×). (

**c**) Microscope image of a square-wave-modulated LPFG (magnification: 100×).

**Figure 6.**Spectrum evolution during LPFG fabrication (period: 560 μm; duty cycle: 50%. Insertion loss from 1200 to 1350 nm increases with the grating length).

**Figure 7.**Transmission spectra of LPFGs with different duty cycles (insertion loss of the LPFG with a duty cycle of 10% is the smallest, and that of the LPFG with a duty cycle of 25% is the highest).

**Figure 8.**Transmission spectra of LPFGs with periods of 560, 1120, and 1680 μm (grating length: 40,320 μm. Resonant intensity decreases with the period, and when the duty cycle is 50%, the second-order harmonic resonance of the grating disappears).

**Figure 9.**Spectrum evolution during LPFG fabrication (insertion loss increases with the grating length).

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**MDPI and ACS Style**

Zhao, X.; Li, H.; Rao, B.; Wang, M.; Wu, B.; Wang, Z.
Spectral Characteristics of Square-Wave-Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser. *Sensors* **2021**, *21*, 3278.
https://doi.org/10.3390/s21093278

**AMA Style**

Zhao X, Li H, Rao B, Wang M, Wu B, Wang Z.
Spectral Characteristics of Square-Wave-Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser. *Sensors*. 2021; 21(9):3278.
https://doi.org/10.3390/s21093278

**Chicago/Turabian Style**

Zhao, Xiaofan, Hongye Li, Binyu Rao, Meng Wang, Baiyi Wu, and Zefeng Wang.
2021. "Spectral Characteristics of Square-Wave-Modulated Type II Long-Period Fiber Gratings Inscribed by a Femtosecond Laser" *Sensors* 21, no. 9: 3278.
https://doi.org/10.3390/s21093278