# Arbitrary-Order Photonic Hilbert Transformers Based on Phase-Modulated Fiber Bragg Gratings in Transmission

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## Abstract

**:**

## 1. Introduction

## 2. Principle and Method

## 3. Design Results and Discussion

#### 3.1. 0.5th-Order Photonic Hilbert Transformer

^{−1}. According to the obtained grating profile in Figure 2a, we calculate the simulated spectrum response of designed PM-FBG employing transfer matrix method and compare it with the ideal SR ${H}_{d1}\left(f\right)$ in Figure 2b. From Figure 2b, we can see that the simulated SR agrees well with the ideal SR in a bandwidth of 4 nm, which means that the PM-FBG we designed could realize the function of a 0.5th-order photonic Hilbert transformer.

#### 3.2. First-Order Photonic Hilbert Transformer

^{−1}. Figure 3b shows the comparison between ideal SR ${H}_{d2}\left(f\right)$ and simulated SR of the designed PM-FBG, which is calculated using the obtained grating profile in Figure 3a and employing the transfer matrix method. Obviously, the simulated SR of our designed PM-FBG agrees well with the ideal SR in a bandwidth of 4 nm.

#### 3.3. 1.5th-Order Photonic Hilbert Transformer

^{−1}, while the grating period variation oscillates in the range of −2.5 nm to 2.2 nm. Figure 4b shows the comparison between ideal SR ${H}_{d3}\left(f\right)$ and simulated SR of the designed PM-FBG. It still can be seen that the simulated SR agrees well with the ideal SR in a bandwidth of 4 nm.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Amplitudes of transmission spectral response ${H}_{d}\left(f\right)$ corresponding to different orders.

**Figure 2.**(

**a**) Grating period (blue line) and coupling coefficient (red line) of the designed 0.5th-order photonic Hilbert transformer (PHT) and (

**b**) comparison between ideal spectrum (dashed red line) and simulated spectrum (blue line).

**Figure 3.**(

**a**) Grating period (blue line) and coupling coefficient (red line) of the designed first-order PHT and (

**b**) comparison between ideal spectrum (dashed red line) and simulated spectrum (blue line).

**Figure 4.**(

**a**) Grating period (blue line) and coupling coefficient (red line) of the designed 1.5th-order PHT and (

**b**) comparison between ideal spectrum (dashed red line) and simulated spectrum (blue line).

**Figure 5.**Temporal responses of the designed phase-modulated fiber Bragg gratings (PM-FBGs): (

**a**) 3 ps-full-width at half-maximum (FWHM) input Gaussian pulse; output pulse of the designed (

**b**) 0.5th-order PHT; (

**c**) first-order PHT; and (

**d**) 1.5th-order PHT.

**Figure 6.**Comparison between the ideal (dashed red line) and simulated (blue line) output pulses for (

**a**) 0.5th-order PHT; (

**b**) first-order PHT; and (

**c**) 1.5th-order PHT.

**Figure 7.**Cross-correlation coefficients between the ideal and numerically obtained temporal output waveforms as a function of the FWHM of the input Gaussian pulse for the three designed PHTs.

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**MDPI and ACS Style**

Li, Y.; Liu, X.; Shu, X.; Zhang, L.
Arbitrary-Order Photonic Hilbert Transformers Based on Phase-Modulated Fiber Bragg Gratings in Transmission. *Photonics* **2021**, *8*, 27.
https://doi.org/10.3390/photonics8020027

**AMA Style**

Li Y, Liu X, Shu X, Zhang L.
Arbitrary-Order Photonic Hilbert Transformers Based on Phase-Modulated Fiber Bragg Gratings in Transmission. *Photonics*. 2021; 8(2):27.
https://doi.org/10.3390/photonics8020027

**Chicago/Turabian Style**

Li, Yanxin, Xin Liu, Xuewen Shu, and Lin Zhang.
2021. "Arbitrary-Order Photonic Hilbert Transformers Based on Phase-Modulated Fiber Bragg Gratings in Transmission" *Photonics* 8, no. 2: 27.
https://doi.org/10.3390/photonics8020027