1.7.3 Propagation along the principal plane *Y Z*

Another special case of light propagation in an anisotropic crystal is that of propagation along a principal plane, such as the *XY, YZ* or *ZX* planes. We start with the propagation vector **k** lying in the *YZ* plane, so that *m _{X}* = 0 and ${m}_{Y}^{2}+{m}_{Z}^{2}=1$. With

*m*= 0, Eqns. 1.55 give

_{X}$\mathcal{A}={n}_{Y}^{2}\text{\hspace{0.17em}}{m}_{Y}^{2}+{n}_{Z}^{2}\text{\hspace{0.17em}}{n}_{Z}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathcal{B}={n}_{Y}^{2}\text{\hspace{0.17em}}{m}_{Z}^{2}+{n}_{X}^{2}\mathcal{A},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathcal{C}={n}_{X}^{2}\text{\hspace{0.17em}}{n}_{Y}^{2}\text{\hspace{0.17em}}{n}_{Z}^{2}.$ |
(1.109) |

The two cases of *n _{X}* <

*n*<

_{Y}*n*and

_{Z}*n*>

_{X}*n*>

_{Y}*n*are considered separately in the next two subsections.

_{Z}1.7.4 k along *YZ* plane, Case 1: *n _{X}* <

*n*<

_{Y}*n*

_{Z}In this case, from 1.56 $\mathcal{D}={n}_{Y}^{2}\text{\hspace{0.17em}}{n}_{Z}^{2}-{n}_{X}^{2}\mathcal{A}$, since $\mathcal{D}$ is defined to be positive and $\mathcal{A}$ takes the values from ${n}_{Y}^{2}$ to ${n}_{Z}^{2}$ as *m _{Y}* goes from 1 to 0. Thus, the possible values of

*n*(from Eqn. 1.57) are

${n}_{}$ |

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