Liu Xin, Shu Xuewen
Appl Opt. 2017 Aug 20;56(24):6714-6719. doi: 10.1364/AO.56.006714.
All-optical fractional-order temporal differentiators with bandwidths reaching terahertz (THz) values are demonstrated with transmissive fiber Bragg gratings. Since the designed fractional-order differentiator is a minimum phase function, the reflective phase of the designed function can be chosen arbitrarily. As examples, we first design several 0.5th-order differentiators with bandwidths reaching the THz range for comparison. The reflective phases of the 0.5th-order differentiators are chosen to be linear phase, quadratic phase, cubic phase, and biquadratic phase, respectively. We find that both the maximum coupling coefficient and the spatial resolution of the designed grating increase when the reflective phase varies from quadratic function to cubic function to biquadratic function. Furthermore, when the reflective phase is chosen to be a quadratic function, the obtained grating coupling coefficient and period are more likely to be achieved in practice. Then we design fractional-order differentiators with different orders when the reflective phase is chosen to be a quadratic function. We see that when the designed order of the differentiator increases, the obtained maximum coupling coefficient also increases while the oscillation of the coupling coefficient decreases. Finally, we give the numerical performance of the designed 0.5th-order differentiator by showing its temporal response and calculating its cross-correlation coefficient.
利用透射光纤布拉格光栅展示了带宽达到太赫兹（THz）值的全光分数阶时间微分器。由于所设计的分数阶微分器是最小相位函数，因此所设计函数的反射相位可以任意选择。作为示例，我们首先设计了几个带宽达到太赫兹范围的0.5阶微分器用于比较。0.5阶微分器的反射相位分别选择为线性相位、二次相位、三次相位和双二次相位。我们发现，当反射相位从二次函数变为三次函数再变为双二次函数时，所设计光栅的最大耦合系数和空间分辨率都会增加。此外，当反射相位选择为二次函数时，所获得的光栅耦合系数和周期在实际中更有可能实现。然后，当反射相位选择为二次函数时，我们设计了不同阶数的分数阶微分器。我们看到，当微分器的设计阶数增加时，所获得的最大耦合系数也会增加，而耦合系数的振荡会减小。最后，我们通过展示其时间响应并计算其互相关系数，给出了所设计的0.5阶微分器的数值性能。
