双曲几何光学：双曲玻璃
Hyperbolic geometrical optics: Hyperbolic glass
摘要：
我们研究了由$n(x,y)=1/y$ $(y>0)$形式的折射率产生的几何光学，其中$y$是$\R^2$中垂直轴在正交参考系中的坐标。因此，我们得到我们所谓的“双曲几何光学“，因为光线轨迹是测地线在庞加莱-罗巴切夫斯基半平面$\HH^2$。然后证明了常位相面是均环，得到了与经典泊松核密切相关的、类似于欧氏平面波的\emph{horocyclic waves}。通过研究Beltrami伪球中的输运方程，我们证明了（i）在$\HH^2$中的整个带状$0<y\leqslant 1$中的流守恒，这是射线轨迹所在的物理感兴趣的有限区域;（ii）轨迹密度的不均匀分布：射线确实朝着水平x轴聚焦，这是$\HH^2$的边界。最后利用正弦-戈登方程详细分析了光线的聚焦和散焦过程。
英文摘要：
We study the geometrical optics generated by a refractive index of the form $n(x,y)=1/y$ $(y>0)$, where $y$ is the coordinate of the vertical axis in an orthogonal reference frame in $\R^2$. We thus obtain what we call "hyperbolic geometrical optics" since the ray trajectories are geodesics in the Poincaré-Lobachevsky half--plane $\HH^2$. Then we prove that the constant phase surface are horocycles and obtain the \emph{horocyclic waves}, which are closely related to the classical Poisson kernel and are the analogs of the Euclidean plane waves. By studying the transport equation in the Beltrami pseudosphere, we prove(i) the conservation of the flow in the entire strip $0<y\leqslant 1$ in $\HH^2$, which is the limited region of physical interest where the ray trajectories lie; (ii) the nonuniform distribution of the density of trajectories: the rays are indeed focused toward the horizontal x axis, which is the boundary of $\HH^2$. Finally the process of ray focusing and defocusing is analyzed in detail by means of the sine--Gordon equation.
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