Triangular Interferometers

• Triangular interferometers are instruments with three 60° arms that exploit equilateral symmetry for unique signal decomposition and null stream creation.
• Their configurations enable robust noise rejection by constructing null streams and decomposing the 3D measurement space into a 2D signal channel and a 1D noise channel.
• They are applied in gravitational-wave detection, photonics, and quantum sensing, offering enhanced sensitivity, redundancy, and reduced computational requirements.

A triangular interferometer is a class of wave-interference instrument in which three arms or measuring elements are arranged at 60° relative angles to form an equilateral triangle, enabling unique detection modes, robust noise suppression, and efficient decomposition of signal spaces. These configurations appear across cutting-edge gravitational-wave observatories, photonic networks, quantum sensors, and atomic physics, with architectures tailored to leverage the inherent geometrical symmetries of a triangle for scientific measurement and information extraction.

1. Triangular Interferometer Architectures

The triangular topology serves as a unifying structural principle, with prominent realizations including:

• Einstein Telescope (ET): Three co-located L-shaped Michelson interferometers, each sharing a vertex and oriented at 60° to each other, with common central and end stations. Each interferometer has arms along unit vectors $u, v, w$, with configurations: (u,v), (v,w), and (w,u) (Wong et al., 2021). This tightly-integrated geometry enables both maximal baseline diversity and robust signal redundancy.
• Triangular Optical Cavities: Triangular mode-cleaning cavities employ three mirrors (usually two flat and one curved) arranged as an isosceles or equilateral triangle, appearing ubiquitously in precision optical systems (Tanioka et al., 2020).
• Triangular Photonic Meshes: Arrays of tunable Mach–Zehnder interferometers (MZIs) in a triangular grid ("Reck" mesh) realize arbitrary unitary transformations for $N \times N$ optical or quantum information processing (Hamerly et al., 2021).
• Space-based GW Detectors: E.g., LISA, Taiji, and TianQin, where three spacecraft in equilateral triangles form synthesized Michelson/TDI measurements. The three arms are functionally equivalent and provide multiple null channels and unique orbital coverage (Seto, 2020).
• Triangular Tricoupler Chips: Integrated photonic devices with three symmetric ports—used in astronomical interferometry for high-throughput, phase-tracking, and nulling purposes (Hansen et al., 2021, Martinod et al., 2021).

2. Signal Spaces, Null Streams, and Decomposition

Triangular architectures naturally partition the measurement space into orthogonal subspaces supporting signal extraction, redundancy, and noise suppression.

• Null Stream Construction: In the ET geometry, summing the strain outputs of all three interferometers exactly cancels the gravitational-wave signal due to the tensorial relation $D_1 + D_2 + D_3 = 0$. The resultant "null stream" $N(t) = s_1(t) + s_2(t) + s_3(t)$ lies in a one-dimensional subspace orthogonal to the physical GW signal (Wong et al., 2021). This enables robust, real-time vetoes of instrumental artifacts and direct noise measurement, as the null stream is uncontaminated by astrophysical signals.
• Projective Decomposition: The 3D observation space splits into a 2D "signal" subspace (carrying all GW information) and a 1D null space. Orthogonal projectors $P_S$ and $P_N$ implement this decomposition, with explicit forms:

$P_N = \frac{1}{3}\begin{pmatrix}1 & 1 & 1\1 & 1 & 1\1 & 1 & 1\end{pmatrix},\quad P_S = I_3 - P_N.$

The rotated basis $(e_1, e_2, e_3)$ aligns the last vector with the null stream (Wong et al., 2021).

• Equivalence to Reduced Likelihoods: Bayesian inference using only the two signal subspace channels yields identical posteriors, evidences, and Bayes factors as using all three strain time series. This justifies dimensionality reduction, with up to 33% memory and $\sim$1.5× computational savings (Wong et al., 2021).

3. Triangular Interferometry: Applications in Gravitational-Wave Detection

Triangular topologies underpin multiple generations of GW observatories:

• Enhanced Sensitivity and Sky Coverage: The three-body geometry eliminates long blind directions, yielding nearly isotropic antenna patterns with RMS sensitivity variation $<10\%$—in contrast to the $\sim50\%$ for a conventional L-shaped Michelson (Liu et al., 2020).
• Redundancy and Null Streams: Internal null-streams formed by explicit linear combinations of outputs allow (i) robust vetoing of glitches, (ii) unbiased noise power spectral density (PSD) estimation even under high-event rates, and (iii) commissioning diagnostics decoupled from astrophysical signals (Wong et al., 2021, Liu et al., 2020).
• Parameter Estimation: Triangular or 3D-pyramid configurations (e.g., the proposed regular triangular pyramid detector) provide multiple independent pattern measurements, shrinking localization uncertainties ($\Delta\Omega$) by $2$–$5\times$ relative to a single Michelson at the same site (Liu et al., 2020).
• Displacement Noise-Free Interferometry (DFI): Equilateral-triangle cavities support decoherence-free subspaces orthogonal to mirror displacement noise directions, enabling quantum-noise-limited strain sensitivity improvements at low frequencies and larger squeezing gains than possible in two-port Michelson architectures (Gefen et al., 2022).
• High-Frequency GW Detection: Triangular resonant loops produce comb-like frequency responses at $f_n = n\,c/(3L)$, where GW-induced phases build up coherently, enabling detection of stochastic gravitational-wave backgrounds up to tens of kHz and beyond, with characteristic narrowband sensitivity unmatched by conventional detectors (Heisig, 15 Oct 2025).

Architecture	Main Capability	Key Null Structure
ET Triangular Network (Wong et al., 2021)	All-sky GW detection, robust null stream	$N(t) = s_1 + s_2 + s_3$
LISA/TianQin Triangle (Seto, 2020)	TDI channels (A/E/T), symmetric sensitivity	$T = (X+Y+Z)/\sqrt{3}$
3D Michelson Pyramid (Liu et al., 2020)	Spherical antenna, null stream across triplet	$n(t) = d_{XY} + d_{YZ} - d_{ZX}$

4. Photonic and Quantum Triangular Meshes

Triangular interferometer concepts extend to general photonic and quantum unitary processors:

• Triangular Meshes: "Reck" meshes of tunable MZIs arranged on a triangular grid synthesize arbitrary $N \times N$ unitaries. Algorithmic stability in configuring large meshes is achieved using the "power-ratio method," based on output power ratios rather than sequential amplitude/phase matching, yielding error scalings $\mathcal{E}_{\rm corr}\sim N\sigma^2$ and robust performance even with large device imperfections (Hamerly et al., 2021).
• Triangular Tricouplers: Integrated photonic chips fabricated with femtosecond-laser waveguide inscription form three-port devices, with symmetric scattering matrices enforcing equal flux splitting and achromatic nulls. These devices enable real-time fringe-tracking and nulling for astronomical interferometry, with 85% throughput and precise phase extraction (Hansen et al., 2021, Martinod et al., 2021).
• Quantum Interferometry and the Tritter: The "tritter" is the quantum analog—a 3D waveguide device performing a true 3-mode Fourier transform. Probing with Fock states yields nonclassical multi-photon fringes, outperforming all classical-coherence quantum Fisher information bounds and enabling multiparameter quantum estimation in a compact, phase-robust geometry (Spagnolo et al., 2013).

5. Triangular Topologies in Classical and Quantum Interference Measurement

Triangular arrangements enable unique measurement invariants through closure relations:

• Closure Phase Invariants: In arrays (radio, optical, or intensity interferometry), the closure phase for a triangle of baselines, $\phi_{123} = \arg(V_{12} V_{23} V_{31})$, is strictly invariant to individual element-based phase fluctuations (Thyagarajan et al., 2020). The geometric "SOS conservation principle" ensures that the triangle's shape, orientation, and size in image space constrain source morphology independent of calibration errors.
• Bispectrum Extraction in Intensity Interferometry: Three-detector HBT ("triangular HBT") accesses the bispectrum $B(u_1, u_2) = V(u_1) V(u_2) V^*(-u_1 - u_2)$, permitting closure-phase-based phase retrieval even in quantum or thermal light with only intensity measurement, albeit with stringent SNR requirements due to $SNR \propto (r\Delta\tau)^{3/2}$ scaling (Malvimat et al., 2013).
• Atomic Triangular Lattices: Ramsey interferometry between Bloch bands in a triangular optical lattice exploits shortcut pulse schemes to create and readout motional-state superpositions, with potential for 2D quantum simulation and precision measurement (Dong et al., 2022).

6. Angular Response, Sensing, and Control in Triangular Cavities

Triangular optical cavities exhibit distinctive sensing and alignment properties:

• Degrees of Freedom and Sensing Matrices: For a cavity formed by three mirrors, only two independent geometric modes (waist translation and tilt) are observable via wavefront sensors; the third (common rotation) is a null direction not coupled to measurable error signals (Tanioka et al., 2020).
• Response Matrix Formalism: The angular response to mirror pitch/yaw tilts is captured by 3×3 response matrices relating mirror fields to observable shifts and WFS outputs, with explicit linear dependence on cavity g-factors and mirror geometry (Tanioka et al., 2020).
• Design Constraints: Validity of analytic models is restricted to small misalignments; commissioning and control loop design must ensure operation within the perturbative regime to guarantee precise modal correction and locking fidelity.

7. Practical Implications, Computational Advantages, and Future Directions

Triangular interferometers afford significant practical and computational benefits:

• Data Reduction: By projecting onto the irreducible signal subspace, storage and likelihood evaluation are reduced by one third for three-channel detectors (Wong et al., 2021).
• Unbiased Noise Estimation: The null stream provides an unbiased estimate of PSD and non-Gaussianity in noise, which is critical for both data quality monitoring and detector commissioning (Wong et al., 2021).
• Generalization to Higher-Order Networks: The formal projection and null-space framework generalizes to any co-located $n$-gon detector networks, including possible four- and six-arm architectures for advanced GW detection and quantum sensing.
• Real-Time Sensing and Robust Nulling: Integrated triangular couplers and photonic chips enable simultaneous real-time extraction of all interference parameters (visibility amplitudes, phases, null depths), with inherent symmetries affording resilience to fabrication and alignment errors (Hansen et al., 2021, Martinod et al., 2021).

Triangular interferometers thus constitute a foundational theme in modern wave and quantum measurement science, with their geometric symmetries yielding structural signal-redundancy, robust noise rejection, and efficient exploitation of higher-order measurement correlations. Ongoing developments in gravitational-wave astronomy, photonic computing, closure-phase spectroscopy, and quantum information continue to extend the utility of triangular architectures into new domains of precision measurement and information processing.