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 \(\sin{\alpha_2} > 1.0\).