finesse.components.beamsplitter.Beamsplitter.ABCD

Beamsplitter.ABCD(from_node, to_node, direction='x', symbolic=False, copy=True, retboth=False, allow_reverse=False)

Returns the ABCD matrix of the beam splitter for the specified coupling.

The matrices for transmission and reflection are different for the sagittal and tangential planes ($M_s$ and $M_t$), as shown below.

Transmission

For the tangential plane (direction = ‘x’),

$$M_t = \begin{pmatrix} \frac{\cos{\alpha_2}}{\cos{\alpha_1}} & 0 \\ \frac{\Delta n}{R_c} & \frac{\cos{\alpha_1}}{\cos{\alpha_2}} \end{pmatrix},$$

and for the sagittal plane (direction = ‘y’),

$$M_s = \begin{pmatrix} 1 & 0 \\ \frac{\Delta n}{R_c} & 1 \end{pmatrix},$$

where $\alpha_1$ is the angle of incidence of the beam splitter and $\alpha_2$ is given by Snell’s law ($n_1\sin{\alpha_1} = n_2\sin{\alpha_2}$). The quantity $\Delta n$ is given by,

$$\Delta_n = \frac{n_2 \cos{\alpha_2} - n_1 \cos{\alpha_1}}{ \cos{\alpha_1} \cos{\alpha_2} }.$$

If the direction of propagation is reversed such that the radius of curvature of the beam splitter is in this direction, then the elements $A$ and $D$ of the tangential matrix ($M_t$) are swapped.

Reflection

The reflection at the front surface of the beam splitter is given by,

$$M_t = \begin{pmatrix} 1 & 0 \\ -\frac{2n_1}{R_c \cos{\alpha_1}} & 1 \end{pmatrix},$$

for the tangential plane, and,

$$M_s = \begin{pmatrix} 1 & 0 \\ -\frac{2n_1 \cos{\alpha_2}}{R_c} & 1 \end{pmatrix},$$

for the sagittal plane.

At the back surface $R_c \rightarrow - R_c$ and $\alpha_1 \rightarrow - \alpha_2$.

See Connector.ABCD() for descriptions of parameters, return values and possible exceptions.

Raises

treTotalReflectionError

If total reflection occurs for the specified coupling - i.e. if $\sin{\alpha_2} > 1.0$.