# Christoffel equation in the polarization variables

## TL;DR

This paper reformulates the Christoffel equation using polarization variables, analyzing the number and nature of solutions for slowness vectors in anisotropic solids, revealing cases with unattainable polarization directions.

## Contribution

It introduces a novel formulation of the Christoffel equation in polarization variables and characterizes the solution set and polarization field properties in triclinic solids.

## Key findings

- Number of solutions varies from 1 to 4 for non-degenerate cases.
- Identifies conditions where polarization fields have holes.
- Discovers finite-size solid angles of unattainable polarization directions.

## Abstract

We formulate the classic Christoffel equation in the polarization variables and solve it for the slowness vectors of plane waves corresponding to a given unit polarization vector. Our analysis shows that, unless the equation degenerates and yields an infinite number of different slowness vectors, the finite nonzero number of its legitimate solutions varies from 1 to 4. Also we find a subset of triclinic solids in which the polarization field can have holes; there exist finite-size solid angles of polarization directions unattainable to any plane wave.