Computing Laser Beam Paths in Optical Cavities: An Approach Based on Geometric Newton Method

Abstract

In the last decade, increasing attention has been drawn to high-precision optical experiments, which push resolution and accuracy of the measured quantities beyond their current limits. This challenge requires to place optical elements (e.g., mirrors, lenses) and to steer light beams with subnanometer precision. Existing methods for beam direction computing in resonators, e.g., iterative ray tracing or generalized ray transfer matrices, are either computationally expensive or rely on overparameterized models of optical elements. By exploiting Fermat’s principle, we develop a novel method to compute the steady-state beam configurations in resonant optical cavities formed by spherical mirrors, as a function of mirror positions and curvature radii. The proposed procedure is based on the geometric Newton method on matrix manifold, a tool with second-order convergence rate, that relies on a second-order model of the cavity optical length. As we avoid coordinates to parametrize the beam position on mirror surfaces, the computation of the second-order model does not involve the second derivatives of the parametrization. With the help of numerical tests, we show that the convergence properties of our procedure hold for non-planar polygonal cavities, and we assess the effectiveness of the geometric Newton method in determining their configurations with high degree of accuracy and negligible computational effort.

Appendix: Discussion and Convergence of Algorithm 1

In Algorithm 1 at each iteration, the Newton Eq. (5) is solved for the function f, then the function $h(x)=\left\| \text {grad}\, f(x)\right\| ^{2}$ is minimized along the computed direction. In this way, we need to compute only Hess f and grad f, avoiding the computation of Hess h, that would require to compute the third derivative of f.

Proposition 1

Algorithm 1 converges to the stationary point $x^{*}$ of the function f with quadratic convergence rate, provided that, in a neighborhood $\mathcal {I}(x^{*})$ of $x^*$, $\text {grad}\, f\ne 0$, $\text {Hess}\, f$ is injective, and the first iterate is $x_{0}\in {\mathcal {I}}(x^{*})$.

Proof

Let x denote a generic algorithm iterate. By hypotheses, the Newton vector $\eta _{x}$, solution of (5), is well defined. The Riemannian gradient of h reads

$$\begin{aligned} \text {grad}\, h=2\text {Hess}\, f\left[ \text {grad}\, f\right] . \end{aligned}$$

(28)

By evaluating the expression $Dh(x)[\eta _{x}]$, we get

$$\begin{aligned} Dh(x)[\eta _{x}]= & {} 2\left\langle \text {grad}\, f(x),\text {Hess}\, f(x)\left[ \text {Hess}\, f(x)^{-1}\left[ -\text {grad}\, f(x) \right] \right] \right\rangle \nonumber \\= & {} -\,2\left\| \text {grad}\, f(x)\right\| ^{2} = -2h(x). \end{aligned}$$

(29)

The sequence $\left\{ \eta _{x_{k}}\right\}$ is gradient related to $\left\{ x_{k}\right\}$. In fact by hypothesis and (28), it holds grad$\, h(x_{k})\ne 0$; therefore, using (29), we get $-2\sup _{\mathcal {I}(x^{*})}h(x_{k})=\sup _{\mathcal {I}(x^{*})}Dh(x_{k})[\eta _{x_{k}}]<0$. By the smoothness of the functional Hess f and of the vector field grad f, since ${\mathcal {I}}(x^{*})$ is a compact set, we can conclude that $\left\{ \eta _{x_{k}}\right\}$ is bounded. Hence Algorithm 1 fits in the framework of Theorem 4.3.1 and Theorem 6.3.2 [13, Chapters 4–6], stating that every accumulation point of $\left\{ x_{k}\right\}$ is a critical point of h, so that the local quadratic convergence holds. $\square$

Note that the Armijo condition (7) for the function h and the direction $\eta _{x}$ can be rewritten as

$$\begin{aligned} h(x)-h\left( y_{k}\right)&<-\sigma \gamma _{k}Dh(x)[\eta _{x}] =2\sigma \gamma _{k}h(x) \end{aligned}$$

(30)

$$\begin{aligned} h\left( y_{k}\right)&>\left( 1-2\sigma \gamma _{k}\right) h(x), \end{aligned}$$

(31)

where $y_{k}=R_{x}(\gamma _{k}\eta _{x})$, $x=x_{k}$, $\eta _{x}=\eta _{x_{k}}$, and k is the iteration number.