Fresnel-Kummer 曲面的精确商与认证双轴折射

中文摘要 AI 辅助

Fresnel 波面支配光在透明双轴晶体中的传播。它是一种特殊的 Kummer 四次曲面，我们对其进行了精确识别。在复数域上，晶体的波面是某个显式亏格二曲线的 Jacobian 的 Kummer 曲面，该亏格二曲线在三个主介电常数的带符号平方根处分枝。该 Jacobian 通过一个核为四阶的同源映射，与两条椭圆曲线的乘积同源。其中一条椭圆曲线承载三个介电常数，另一条承载光轴角。物理族在具有额外对合的亏格二曲线轨迹中是 Zariski 稠密的，其自同构分层是显式的。该识别实例化了一个任务感知商，它识别相差一个干扰变换的参数，并携带不变坐标和显式分层。对于双轴晶体，介电常数的两个比率构成波面在旋转和重缩放下的完全不变量，四个实节点以闭式给出。在界面处，候选透射波是精确四次多项式的根。其实根计数、根序和重根事件由精确代数谓词决定，并且在本文的通用单节点遭遇中，其判别式二阶消失。浮点求解器在光轴附近丢失前向透射模式，而认证求解器不会。在精确等价类上，基于商坐标的学习模型比基于原始张量的模型更不变且更准确，而学习到的根计数谓词在光轴附近失败。

英文摘要

The Fresnel wave surface governs the propagation of light in a transparent biaxial crystal. It is a special Kummer quartic, and we identify it exactly. Over the complex numbers the wave surface of a crystal is the Kummer surface of the Jacobian of an explicit genus-two curve branched at the signed square roots of the three principal permittivities. This Jacobian is isogenous, by an isogeny with kernel of order four, to a product of two elliptic curves. One elliptic curve carries the three permittivities, and the other carries the optic-axis angle. The physical family is Zariski dense in the locus of genus-two curves with an extra involution, and its automorphism strata are explicit. The identification instantiates a task-aware quotient, which identifies parameters that differ by a nuisance transformation and carries invariant coordinates and explicit strata. For biaxial crystals, two ratios of the permittivities form a complete invariant of the wave surface up to rotation and rescaling, and the four real nodes are given in closed form. At an interface the candidate transmitted waves are the roots of a quartic of exact degree four. Its real-root count, root order, and repeated-root events are decided by exact algebraic predicates, and along the generic single-node encounters of the paper its discriminant vanishes to second order. Floating-point solvers drop forward transmitted modes near the optic axes, and the certified solver does not. On exact equivalence classes, learned models on quotient coordinates are invariant and more accurate than models on raw tensors, while learned root-count predicates fail near the optic axes.

发表机构

• Oakland University（奥克兰大学）

机构由 AI 辅助整理，请以论文原文为准。