Bispectral optical cavity based on twin metamirrors

. In this work, we introduce a new bispectral optical cavity concept for which we design twin pairs of highly re ﬂ ective, ultra-low noise metamirrors. Metasurfaces, arti ﬁ cial structures composed of periodic or quasi-periodic arrays of nanostructures, offer unprecedented control over light properties, paving the way for new applications in areas from high-precision optical metrology to quantum science. Custom phase and an ultra-high re ﬂ ection coef ﬁ cient make these metasurfaces an ideal candidate to surpass traditional multilayer mirrors as metamirrors in precision interferometry, particularly by also minimizing thermal noise. The focusing metamirrors designed in this study expect to re ﬂ ect 99.95% and 99.96% of the incoming light at both, 1064 nm and 1550 nm wavelength. Their planar counterparts even reach theoretical re ﬂ ectivities of 99.9999% (1064 nm) and 99.9995% (1550 nm). These specialized metamirrors enable bispectral low-noise optical cavities, which would reduce the number of cavities in optical experiments or could be used as a versatile transfer cavity for frequency locking.


Introduction
The field of high-precision optical metrology heavily relies on ultrastable laser systems, specifically their unmatched frequency stability.This ability to provide minimal fluctuations in frequency over time has opened the door to investigate the most fundamental laws of nature, such as variations of the fine structure constant [1] or a test of special relativity [2], just to name a few examples.Furthermore, the intricate search for gravitational waves is based on interferometers that use ultrastable lasers [3].
However, the current limit of these systems lies in the Brownian noise of the mirror coatings [4].Microstructured surfaces, i.e. metasurfaces, promise a low-noise alternative [5] to conventional Bragg mirrors by introducing near-field effects as the driving mechanism to achieve high reflectivities [6].Metamirrors not only allow for the modulation of the amplitude of the light field, but also for the control of other properties such as polarization and phase.The latter is essential since stable cavities require at least one mirror to have focusing abilities [7].Metaatoms, i.e. the building blocks of the metasurface, are the key to this high level of light control.We use a pathfinder design approach to render metamirror pairs that can be employed in a cavity.

Pathfinding our way through the design space
We begin by starting with a cross-shaped metaatom, as shown in Fig. 1 which is defined by four structural parameters: L x , L y , h and P. While we use the first two * e-mail: liam.shelling-neto@tu-braunschweig.de Figure 1.Schematic of the cross-shaped metaatom with variable lateral height and width L x and L y .The material system is chosen to be amoprhous silicon (a-Si) on Silica (SiO 2 ).The structural height of both mirrors is fixed to 395 nm, while their periods differ with 555 nm and 893 nm, for 1064 nm and 1550 nm respectively.
parameters to control phase and amplitude, the other two parameters, height h and period P, need to be fixed for both mirrors.For manufacturing reasons, both mirrors have to share the same height but can have different periods.This can be achieved using a black-box optimizer, such as Bayesian Optimization [8], which maximizes the sum of the reflectivities at both wavelengths.We then simulate pairs of L x and L y with RCWA [9] to obtain a dataset of design-response-pairs, where each pair represents a metaatom in which a specific choice of L x and L y yields a value for reflectivity and phase.
Ultrastable optical cavitities require at least one mirror to have focusing abilities.The other one, however, can have a flat phase profile.Therefore, we can first determine a metaatom design for a flat mirror by selecting the best parameter combination based only on reflectivities for both wavelengths and ignoring their phases.The theoretical reflectivity is in this case close to 1.For the focusing metamirrors, we then set a threshold of 99.9 % for reflectivity to to filter the dataset.We choose this threshold such that connected high-reflectivity areas still exist in the dataset (see Fig. 2).We select the area with the highest phase difference to enable a pathfinding algorithm to collect appropriate designs along its path.As the pathfinding algorithm traverses the design space step-by-step, the structural parameters of neighboring metaatoms do not change significantly.This is essential to account for neighboring coupling effects, since each metaatom is simulated under the assumption of periodic boundary conditions.The resulting phase ramp is depicted in Fig. 2. It can be seen that the metamirror, while being physically flat, mimics the phase profile of a parabolic mirror.Another advantage of metamirrors now becomes apparent.While conventional mirrors need to be super-polished to achieve an extremely flat phase profile, metamirrors allow for the design of large focal lengths with ease.In our case, we choose a focal length of 6 m.
We will use the designed metamirror twins to experimentally realize an optical cavity for 1064 nm and 1550 nm.Moreover, by increasing the parameter space, we aim to achieve even higher reflectivities to eventually reach the regime of ultrastable cavities.by

Figure 2 .
Figure 2. Phase and reflectivity maps with a threshold of 99.9 % applied to the dataset.The black dotted line represents the path taken by the path finding algorithm.

Figure 3 .
Figure 3. Radial phase profile of the metamirror pair for 1064 nm and 1550 nm.Both mirrors yield theoretical reflectivities of > 99.9 % at a size of 3 × 3 mm and a focal length of 6 m.The inset shows the stepwise phase profile caused by the discrete metaatoms.