Complex beam parameter

The complex beam parameter is usually used in ray transfer matrix analysis, which allows the calculation of the beam properties at any given point as it propagates through an optical system, if the ray matrix and the initial complex beam parameter is known. This same method can also be used to find the fundamental mode size of a stable optical resonator.

Given the initial beam parameter, q_i, one can use the ray transfer matrix of an optical system, ${\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}}$, to find the resulting beam parameter, q_f, after the beam has traversed the system:^[1]

${\displaystyle q_{f}={\frac {Aq_{i}+B}{Cq_{i}+D}}}$.

It is often convenient to express this equation in terms of the reciprocals of q:^[1]

${\displaystyle {1 \over q_{f}}={\frac {C+D/q_{i}}{A+B/q_{i}}}}$.

The effect of propagation in free space is just that of adding the travelled axial distance Δz to the complex beam parameter:^[2]

${\displaystyle q_{f}=q_{i}+\Delta z}$.

For simple astigmatic fundamental Gaussian beams,^[3] the q- parameters for the tangential and sagittal planes are independent. This is no longer true if those planes do not coincide with the principal direction of the surface on which the beam impinges; that case is called general astigmatism.^[3] Formulas for an incidence angle θ_i were derived in Massey and Siegman's 1969 paper.^[4]

For reflection, the ABCD matrices read:^[5]

${\displaystyle {\begin{aligned}M_{t}&={\begin{pmatrix}1&0\\{\frac {2}{R_{I}\cos \theta _{i}}}&1\end{pmatrix}},&M_{t}&={\begin{pmatrix}1&0\\{\frac {2\cos \theta _{i}}{R_{S}}}&1\end{pmatrix}}.\end{aligned}}}$

The ones for refraction are:^[6]

${\displaystyle {\begin{aligned}M_{t}&={\begin{pmatrix}{\frac {\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}{n_{r}\cos \theta _{i}}}&0\\{\frac {\cos \theta _{i}-{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}{R_{I}\cos \theta _{i}{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}}&{\frac {\cos \theta _{i}}{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}\end{pmatrix}},&M_{t}&={\begin{pmatrix}1&0\\{\frac {\cos \theta _{i}-{\sqrt {n_{r}^{2}-\sin ^{2}\theta _{i}}}}{R_{S}n_{r}}}&{1 \over n_{r}}\end{pmatrix}}.\end{aligned}}}$

To find the complex beam parameter of a stable optical resonator, one needs to find the ray matrix of the cavity. This is done by tracing the path of beam in the cavity. Assuming a starting point, find the matrix that goes through the cavity and return until the beam is in the same position and direction as the starting point. With this matrix and by making q_i = q_f, a quadratic is formed as:

${\displaystyle C{q_{f}}^{2}+(D-A)q_{f}-B=0}$.

Solving this equation gives the beam parameter for the chosen starting position in the cavity, and by propagating, the beam parameter for any other location in the cavity can be found.

• Kochkina, Evgenia (2013). Stigmatic and astigmatic Gaussian beams in fundamental mode: impact of beam model choice on interferometric pathlength signal estimates (PhD). Gottfried Wilhelm Leibniz Universität Hannover.