标题： PT对称光学结构的解析结果 显示英文标题

标题： Analytic Results for a PT-symmetric Optical Structure

摘要： 光在具有复折射率的介质中传播，其中增益和损耗被设计为$PT$对称，具有许多显著特性。 特别是通常的幺正性关系不成立，因此反射系数可以大于一，并且一般来说，对于左向或右向入射并不相同。 在形式为$v(x)=v_1\cos(2\beta x)+iv_2\sin(2\beta x)$的光学势类中，情况$v_2=v_1$特别引人注目，因为它位于$PT$对称性破缺的边界上。 最近 Lin 等人的论文中已经证明，在这种情况下，具有“单向不可见性”的特性，而在另一方向传播时，反射系数大大增强，与$L^2$成正比，其中$L$是介质在传播方向上的长度。 对于这种势，我们展示了如何获得各种透射和反射系数的解析表达式，这些表达式以修改后的贝塞尔函数的非常简洁的形式表示。 虽然我们的数值结果与 Lin 等人的结果非常吻合，但我们发现不可见性在幅度或相位上并不是完全精确的。 作为我们公式的测试，我们展示了它们恒等满足适用于$PT$对称势的修正版幺正性。 我们还研究了对于波包而非平面波，增强传输是如何发生的，发现现在增强是通过脉冲长度$O(L)$的增加，而不是振幅的增加而产生的。 显示英文摘要

摘要： Propagation of light through media with a complex refractive index in which gain and loss are engineered to be $PT$ symmetric has many remarkable features. In particular the usual unitarity relations are not satisfied, so that the reflection coefficients can be greater than one, and in general are not the same for left or right incidence. Within the class of optical potentials of the form $v(x)=v_1\cos(2\beta x)+iv_2\sin(2\beta x)$ the case $v_2=v_1$ is of particular interest, as it lies on the boundary of $PT$-symmetry breaking. It has been shown in a recent paper by Lin et al. that in this case one has the property of "unidirectional invisibility", while for propagation in the other direction there is a greatly enhanced reflection coefficient proportional to $L^2$, where $L$ is the length of the medium in the direction of propagation. For this potential we show how analytic expressions can be obtained for the various transmission and reflection coefficients, which are expressed in a very succinct form in terms of modified Bessel functions. While our numerical results agree very well with those of Lin et al. we find that the invisibility is not quite exact, in amplitude or phase. As a test of our formulas we show that they identically satisfy a modified version of unitarity appropriate for $PT$-symmetric potentials. We also examine how the enhanced transmission comes about for a wave-packet, as opposed to a plane wave, finding that the enhancement now arises through an increase, of $O(L)$, in the pulse length, rather than the amplitude.