When a monochromatic electromagnetic plane-wave arrives at the flat interface between its transparent host (i.e., the incidence medium) and an amplifying (or gainy) second medium, the incident beam splits into a reflected wave and a transmitted wave. In general, there is a sign ambiguity in connection with the \(k\)-vector of the transmitted beam, which requires at the outset that one decide whether the transmitted beam should grow or decay as it recedes from the interface. The question has been posed and addressed most prominently in the context of incidence at large angles from a dielectric medium of high refractive index onto a gain medium of lower refractive index. Here, the relevant sign of the transmitted \(k\)-vector determines whether the evanescent-like waves within the gain medium exponentially grow or decay away from the interface. We examine this and related problems in a more general setting, where the incident beam is taken to be a finite-duration wavepacket whose footprint in the interfacial plane has a finite width. Cases of reflection from and transmission through a gainy slab of finite-thickness as well as those associated with a semi-infinite gain medium will be considered. The broadness of the spatiotemporal spectrum of our incident wavepacket demands that we develop a general strategy for deciding the signs of all the \(k\)-vectors that enter the gain medium. Such a strategy emerges from a consideration of the causality constraint that is naturally imposed on both the reflected and transmitted wavepackets.

当单色电磁平面波到达其透明主体（即入射介质）和放大（或增益）第二介质之间的平坦界面时，入射光束分裂为反射波和透射波。一般而言，存在与发射光束的\(k\)-向量相关的符号歧义，这要求在一开始就决定发射光束在从界面后退时是增长还是衰减。在从高折射率的电介质以大角度入射到低折射率的增益介质的情况下，这个问题已经被提出并得到了最突出的解决。这里，传输\(k\)的相关符号-vector 确定增益介质内的类渐逝波是远离界面呈指数增长还是衰减。我们在更一般的设置中检查这个和相关问题，其中入射光束被视为有限持续时间的波包，其在界面平面中的足迹具有有限的宽度。将考虑从有限厚度的增益板反射和透射的情况以及与半无限增益介质相关的情况。我们的事件波包的时空谱的广泛性要求我们制定一个通用的策略来决定所有\(k\)- 进入增益介质的向量。这种策略源于对自然施加在反射波包和传输波包上的因果关系约束的考虑。