# Shortcuts to Adiabaticity for Optical Beam Propagation in Nonlinear Gradient Refractive-Index Media

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## Abstract

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## 1. Introduction

## 2. Variational Approximation Method

## 3. Effective Potential and Adiabatic Reference

## 4. Inverse Engineering for Fast Compression

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The potential energy $V\left(a\right)$ for a fictitious particle with unit mass as the function of the beam width a. Here the parameters are ${\alpha}^{2}\left(0\right)=0.3$ and $P=1$ (solid red), $P=0.1$ (dashed black), and $P=10$ (dash-dotted blue).

**Figure 2.**The evolution of beam width designed by STA (solid red) and adiabatic protocols (dashed blue), where the initial and final values are fixed by Equation (14). Parameters: $a\left(0\right)={a}_{i}=1$ and $a\left({z}_{f}\right)=0.5$ are the same for STA and adiabatic protocols, but the propagation distance ${z}_{f}=306$ (adiabatic) and ${z}_{f}=5$ (STA) are different. $P=1$ and the other parameters are the same as those in Figure 1.

**Figure 3.**Guiding coefficient, ${\alpha}^{2}\left(z\right)$, relevant to the parabolic profile of refractive index, for designed STA and adiabatic protocols, when ${z}_{f}=5$ (solid red), ${z}_{f}=0.6$ (dot-dashed black) for STA protocols are compared with adiabatic references ${z}_{f}=306$ (dashed blue).

**Figure 4.**Density plot of effective potential, ${\alpha}^{2}\left(z\right){x}^{2}$, corresponding to the parabolic profile of the refractive index, where the adiabatic reference (

**a**) and STA protocol (

**b**) are presented with the same parameters as those in Figure 2.

**Figure 5.**Beam propagation by using the split-step Fourier method, for free space without parabolic refractive index (

**a**), adiabatic reference (

**b**), and STA protocol (

**c**). All the parameters are the same as those in Figure 2.

**Figure 6.**Fidelity versus the propagation distance ${z}_{f}$ for adiabatic reference (blue dotted) and STA protocol (red dotted). All parameters used here are the same as those in Figure 2.

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**MDPI and ACS Style**

Kong, Q.; Ying, H.; Chen, X. Shortcuts to Adiabaticity for Optical Beam Propagation in Nonlinear Gradient Refractive-Index Media. *Entropy* **2020**, *22*, 673.
https://doi.org/10.3390/e22060673

**AMA Style**

Kong Q, Ying H, Chen X. Shortcuts to Adiabaticity for Optical Beam Propagation in Nonlinear Gradient Refractive-Index Media. *Entropy*. 2020; 22(6):673.
https://doi.org/10.3390/e22060673

**Chicago/Turabian Style**

Kong, Qian, Huimin Ying, and Xi Chen. 2020. "Shortcuts to Adiabaticity for Optical Beam Propagation in Nonlinear Gradient Refractive-Index Media" *Entropy* 22, no. 6: 673.
https://doi.org/10.3390/e22060673