Ground-Based Interferometers and Detector Response

A ground-based interferometer converts tidal spacetime strain into an optical phase, then into a calibrated digital time series. No single transfer function captures the whole instrument: arm geometry projects polarization, cavities store light, finite travel time adds frequency dependence, control and calibration map physical motion to data, and a network adds inter-site phase. This chapter builds that chain from one Michelson to a versioned observing network.

Helpful background. The Wave Equation and Characteristics supplies null propagation and Fourier response. Gravitational Radiation and Waveforms supplies tensor polarizations and strain. Probability and Inference for Gravity supplies likelihood and uncertainty concepts used in network interpretation.

Enter this chapter

Two routes run through the material.

The response route asks how a tensor wave becomes a measured scalar:

$$h_{ij}^{\rm TT} \longrightarrow \delta T_{u,v} \longrightarrow \delta\phi \longrightarrow d_{\rm err} \longrightarrow h_{\rm calibrated}.$$

It begins with the bare Michelson, adds resonant optical storage and finite-arm propagation, and ends at the time-dependent calibration response.

The geometry route asks how direction and polarization enter the data:

$$(h_+,h_\times,\boldsymbol n) \longrightarrow F_I^A \longrightarrow \{d_I\}_{I=1}^{N} \longrightarrow \text{timing and polarization constraints}.$$

It begins with a single detector tensor and ends with coherent network information. The routes meet because detector geometry, optical response, calibration phase, and site delay all multiply or combine in the same predicted data stream.

Throughout the chapter, $\boldsymbol n$ points toward the source and the wave propagates along $-\boldsymbol n$. Frequency-domain phases use the site-wide Fourier convention. A different convention is equally valid only if every propagation, delay, response, and waveform phase is transformed consistently.

The six steps

1. Michelson Interferometer Response derives the round-trip timing and phase difference, defines the normalized detector tensor, and explains why a Michelson measures differential rather than common arm motion.
2. Fabry–Pérot Arms and Optical Cavities replaces a single pass by a coherent geometric series, distinguishing free spectral range, finesse, storage time, and the low-frequency cavity pole.
3. Antenna Patterns and Polarization Response contracts the detector tensor with plus and cross polarization tensors, derives the sky patterns and nulls, and identifies single-detector degeneracies.
4. Finite-Arm and Frequency-Dependent Response integrates the wave phase along the photon path and shows when a real, frequency-independent detector tensor ceases to be adequate.
5. Detector Networks and Triangulation combines site delays, differently oriented antenna patterns, and coherent phase to localize and reconstruct a common signal.
6. Instrument Configurations and Observing States separates reference design, installed hardware, operating state, calibration, and analysis-specific data selection.

The order is deliberate. Antenna patterns use the long-wavelength detector tensor first because the geometry is transparent there. Finite-arm response then promotes the antenna factor to a complex, frequency-dependent quantity. Network response uses either version, depending on the required accuracy.

One response, several physical scales

For an equal-arm instrument of length $L$, a useful schematic response is

$$\widetilde d_I(f) =\sum_A \mathcal C_I(f;t) F_I^A(f,\boldsymbol n,\psi;t) e^{-2\pi if\boldsymbol n\cdot\boldsymbol x_I/c} \widetilde h_A(f) +\widetilde n_I(f).$$

Here $F_I^A$ contains geometric projection and, when required, finite-arm averaging; $\mathcal C_I$ represents the optical, sensing, control, and calibration chain appropriate to the chosen data channel. The detector-tensor and finite-light-time factors are developed in Romano and Cornish 2017, §5.2, PDF, while the coupled optical and control architecture is documented in LIGO Scientific Collaboration 2015, §§2–4, PDF. The factors probe distinct scales:

Scale	Dimensionless control parameter	Physical effect
Arm light travel	$2\pi fL/c$	Phase averaging and finite-arm nulls
Optical storage	$f/f_{\rm pole}$	Cavity amplitude and phase response
Site separation	$2\pi fB/c$	Inter-detector timing phase
Earth rotation	observation duration / sidereal day	Time-varying sky projection
Configuration drift	analysis duration / response-validity interval	Time dependence of calibration and state

The scales must not be collapsed into one generic “detector response.” For example, the small-arm approximation may be excellent while the inter-site phase is essential, and a cavity pole may matter far below a bare arm’s first free-spectral-range null.

What an observation must carry with it

Stable equations can be derived once; observational inputs must be pinned. A reproducible detector prediction identifies the detector geometry, epoch, strain product, calibration version and uncertainty, observing and quality segments, frequency range, sample rate, processing choices, and software environment. The public-data boundary in this chapter is the GWOSC documentation available on 2026-08-18. Any later configuration, release, or caveat must be checked against a newer official record.

Three kinds of evidence should agree:

• analytic checks: dimensional consistency, low-frequency limits, symmetry, and causal delay bounds;
• controlled calculations: numerical path integration, synthetic polarization injections, and coherent delay recovery under refined sampling;
• observational metadata: versioned response, state, and data-quality records attached to the analyzed interval.

Agreement of a waveform with one strain channel does not validate the full chain. A network calculation should repeat under calibration uncertainty and reasonable PSD or conditioning changes, while a configuration comparison should distinguish documented hardware changes from correlations in measured noise.

Review the chapter

Consider a short signal observed by three detectors. Its frequencies satisfy $fL/c\ll1$ for every arm but $fB/c\sim1$ for inter-site baselines. One detector has an antenna response close to a polarization null, and a second detector’s calibration phase has a quoted uncertainty.

1. Which response terms may be approximated as frequency independent?
2. Which phase terms must be retained for localization?
3. Why is a weak signal in the near-null detector not evidence against the candidate?
4. How should calibration uncertainty enter the analysis?

Solution

Because $fL/c\ll1$, each bare arm transfer can be replaced by its long-wavelength limit, so the geometric detector tensor is frequency independent over the signal band. An optical or calibration transfer cannot be discarded from that inequality alone: its own pole and validity must be checked.

The site phases $e^{-2\pi if\boldsymbol n\cdot\boldsymbol x_I/c}$ must be retained because $fB/c\sim1$. They encode the arrival-time differences used in triangulation.

The expected strain is $F_I^+h_++F_I^\times h_\times$. Near a response null, a real signal can be subthreshold even when other detectors observe it strongly. The likelihood must use that predicted weak response and the detector’s actual noise rather than demand equal visibility.

Finally, the calibration phase should be represented by its released frequency-dependent nuisance model or samples and propagated coherently. Fixing it at the nominal value can make timing and sky regions artificially narrow.

References

• Gravitational Wave Open Science Center. “API Documentation.” Version available 2026-08-18. Official documentation.
• LIGO Scientific Collaboration. “Advanced LIGO.” Classical and Quantum Gravity 32 (2015): 074001. DOI. Open PDF.
• Romano, Joseph D., and Neil J. Cornish. “Detection Methods for Stochastic Gravitational-Wave Backgrounds: A Unified Treatment.” Living Reviews in Relativity 20 (2017): 2. DOI. Open PDF.