Next-generation interferometry with gauge-invariant linear optical scatterers

TL;DR

The paper addresses the limitations of traditional interferometers by introducing gauge-invariant, higher-dimensional linear-optical scatterers as building blocks. It develops a formal framework for gauge symmetries and classifies U(d) scatterers (U1, U2, U3, U4), emphasizing Grover coins and Y couplers as versatile, symmetry-driven devices. It then demonstrates how replacing conventional components with these multiport scatterers in Michelson, Sagnac, and Fabry-Pérot geometries yields enhanced phase sensitivity, multi-parameter readouts, and tunable effective finesse, including experimental implementations of Grover-Michelson and analyses of Grover-Sagnac and higher-dimensional Fabry-Pérot. The results indicate significant potential for integrated photonic metrology and multi-axis sensing, with practical pathways via metasurfaces, MMIs, and nanophotonic designs, while noting challenges from losses and non-idealities in real devices.

Abstract

Measurement technology employing optical interference phenomena such as a fringe pattern or frequency shift has been evolving for more than a century. The systems are being designed better, and their components are being built better. But the major components themselves hardly change. Most modern interferometers rely on the same conventional set of components to separate the electromagnetic field into multiple beams, such as plate optics and beam-splitters. This naturally limits the design scope and thus the potential applicability and performance. However, recent investigations suggest that incorporating novel, higher-dimensional linear-optical splitters in interferometer design can lead to several improvements. In this work, we review the underlying theory of these novel optical scatterers and some demonstrated configurations with enhanced resolution. The basic principles of optical interference and optical phase sensing are discussed in tandem. Emphasis is placed on both familiar and unfamiliar scatterers, such as the maximally-symmetric Grover multiport, whose actions are left unchanged by certain gauge transformations. These higher-dimensional, gauge-invariant multiports embody a new class of building blocks which can tailor optical interference for metrology in unconventional ways.

Figures (21)

• Figure 1: Comparison of some common symmetric multiports. The balanced beam-splitter (a) produces two transmitting beams of equal, 50% intensity for any single input, while the four-port Grover coin (b) produces four outgoing beams of 25% intensity, with one of those being a back-reflection. The symmetric Y coupler (c) produces two equally intense beams when input to port 1, with no back-reflection. However, when input to port 2, a 25% back-reflection occurs, while 50% of the energy returns to port 1 and 25% scattering to port 3. Due to the underlying symmetry, input to port 3 is the same as input to port 2. Symmetry, in this context, is not just in the sense of having equal probabilities. The full mathematical description of the symmetries a scattering device can possess is described in Section \ref{['sec:symm']}. These higher-dimensional devices generalize traditional interferometric configurations; some of the examples to be discussed in detail later is shown are Fig. \ref{['fig:comparison02']}.
• Figure 2: Examples of traditional interferometers (a) and their corresponding multiport generalizations (b). The beam-splitters in the Michelson and Sagnac configurations are replaced with the higher-dimensional Grover coin, while the two-dimensional plate optics forming the Fabry-Pérot interferometer are replaced with three-dimensional Y-couplers. In each of the multiport interferometers in case (b), the increased dimensionality combined with back-scattering behavior creates a new pair of coupled resonators. All of the pictured systems, along with other multiport interferometers, will be presented in detail in Section \ref{['sec:interferometry']}.
• Figure 3: A generic, lossless linear-optical scattering device. This $d$-port device is mathematically represented by a $d\times d$ unitary scattering matrix $U$. Port $j$ is identified with two spatial modes, one for each direction of propagation. The scatterer converts an input optical state $|\psi_{\text{in}} \rangle$ to the output state $|\psi_{\text{out}} \rangle = U| \psi_{\text{in}} \rangle$. This scattering action redistributes the optical energy in the input state. The arrows represent the flow of the associated state amplitudes. For clarity the counter-propagating arrows at the same port are drawn spatially separated, but in reality, the counter-propagating energy flow is perfectly collinear in each spatial mode. Scattering devices like $U$ can be viewed as nodes of a general graph. Depictions like the above and more traditional nodal depictions such as those in Fig. \ref{['fig:equiv']} will be used interchangeably. A graph of scattering devices is generally called an interferometer.
• Figure 4: Geometric gauge transformations involve relabeling the field modes identified with the ports of a scattering device $U$. The indices $j$ of the mode operators undergo a permutation transformation $p(j)$, which transforms the scattering matrix $U$ to $PUP^T$, where $P$ is the permutation matrix associated with the permutation function $p$. Permuting the port labels is also equivalent to applying geometric transformations to the scattering device while leaving the labels themselves fixed. Any example is shown here for a generic three-port scattering device, where exchange of two labels equates to reflection about the line between them. The edge colors at the red and blue ports are used to signify the orientation of this pointlike scatterer $U$. General permutations $P$ in higher dimensions will enact reflections and/or rotations in higher-dimensional spaces.
• Figure 5: Two equivalent scattering matrix conventions. In convention (a) the ingoing and outgoing modes are associated with the same port, producing a $d\times d$ matrix $U$. In convention (b) these sets of modes are kept distinct, and assigned to different ports. This produces the $2d \times 2d$ block matrix form in Eq. (\ref{['eq:dirspecific']}). The ports are abstract entities, so their identification with a single mode or pair of counter-propagating modes is immaterial. Nevertheless, certain, so-called block feed-forward scattering matrices expressed in the unbiased convention mimic the block form, and can be compactified further, as in Eq. (\ref{['eq:compact']}).