Conerefringent element

also: CRE, pl. CREs

CRE is a crystal element cut along an optical axis of biaxial crystal. The two opposite sides of the cuts, named bases, are processed to admit the light to pass through.

It is convenient to describe the CRE by its Lambda-parameter and stiff attached to it axis, denoted Y-axis.

Consider the unit eigenvectors $\{
				\mbox{\boldmath$\ \varepsilon_{1} $}, \mbox{\boldmath$\varepsilon_{3}
				$} \} $ associated with the eigenvalues $\varepsilon_{1} < \varepsilon_{2} <
				\varepsilon_{3}$ of crystal dielectric tensor $\mbox{\boldmath$\varepsilon$}$. Without loss of generality, suppose the CRE is cut along the optical axis with vector $\mathbf{O_{1}}$. Denote by $\sigma_{1},
				\sigma_{3}$ the signs of the scalar products $( \mathbf{O_{1}} \cdot
				\mbox{\boldmath$\varepsilon_{1}$})$, $( \mathbf{O_{1}} \cdot \mbox{\boldmath$\varepsilon_{3}$})$, respectively. The two degrees of freedom for the directions of the vectors $\{ \mbox{\boldmath$
				\varepsilon_{1} $}, \mbox{\boldmath$\varepsilon_{3} $} \} $ and $\mathbf{O_{1}}$ generate four combinations $\{\sigma_{1},
				\sigma_{3}\}:\{+, +\}$, $\{+, -\}$, $\{-, +\}$ and $\{-, -\}$. The two combinations with $\sigma_{1} =
				\sigma_{3}$ determine uniquely the vector $\{x, y, z\}$.

Define the direction of the Y-axis along the unit pseudovector $\{x, y, z\}$.

Let $\{x, y, z\}$ is a laboratory Cartesian coordinate system, where the components of above vectors are expressed. If we change the handedness of the system to receive $\{x', y', z'\}$, by plane reflection $\{x', y', z' = -z\}$ or central inversion $\{x' = -y', y' = -y, z' = -z\}$, then for the primed system the direction of the Y-axis changes to the opposite one.

To remove this ambiguity let us associate with CRE right-hand coordinate system {X, Y, Z} . Now we can even mark on each CRE the direction of the Y-axis, as shown with red below. Define the coordinate system in the next manner:

1) The Z-axes is always along the momentum p of the incident photon. In the case of photon beam the Z-axis is along the gravity center of the intensity cross sectional distribution of the incident beam;

2) The X-axis is determined by the convention of the rifght handedness;

3) The plane {X, Y, 0} coincides with the isomomentum plane of the passed through CRE beam.

Image image005

An experimental setup is presented in the figure for two cascaded CREs. In the symbolic picture (in right) the arrows on both CREs show the orientations of their Y-axes seen looking against the beam propagation. For the picture above we can say that the second CRE is rotated relative to the first CRE by angle $\omega$ in CCW direction. The term “CCW” is unambiguously defined for the right-hand coordinate system.


May 24, 2006
Author: T Kirilov