Finite-Arm and Frequency-Dependent Response

The constant Michelson detector tensor assumes that every part of a photon path samples essentially the same gravitational-wave phase. A finite arm instead measures a path average at emission, reflection, and reception. The result is a complex response that depends on frequency and source direction, with a sinc suppression and a propagation phase. This page derives that response for a bare equal-arm Michelson; optical storage and calibration remain separate transfer functions.

Required background. The Wave Equation and Characteristics supplies null propagation, plane-wave phase, and Fourier modes.

Helpful background. Gravitational Radiation and Waveforms fixes the tensor strain convention. Probability and Inference for Gravity becomes useful when response uncertainty is propagated through a likelihood.

A wave sampled along a null path

Let $\boldsymbol n$ point from the beam splitter toward the source, so a plane wave propagates along $-\boldsymbol n$. Use the Fourier pair

$$h_{ij}(t,\boldsymbol x) =\int_{-\infty}^{\infty}df\, \widetilde h_{ij}(f) e^{-2\pi i f(t+\boldsymbol n\cdot\boldsymbol x/c)}.$$

This realizes the site’s $e^{-i\omega t}$ convention. Consider an arm of length $L$ along $\boldsymbol u$, and define

$$x=\frac{fL}{c}, \qquad \mu=\boldsymbol n\mathbin{\cdot}\boldsymbol u, \qquad \operatorname{sinc}z=\frac{\sin z}{z}.$$

To first order in $h_{ij}$, the round-trip timing perturbation can be evaluated on the unperturbed photon path. If the photon returns to the beam splitter at time $t$, use

$$\begin{array}{lll} \text{outbound:}&t_{\rm out}(s)=t-2L/c+s/c,&\boldsymbol x_{\rm out}(s)=s\boldsymbol u,\\ \text{return:}&t_{\rm ret}(s)=t-L/c+s/c,&\boldsymbol x_{\rm ret}(s)=(L-s)\boldsymbol u, \end{array}$$

for $0\le s\le L$. The first-order null condition gives

$$\delta T_{\boldsymbol u}(t) =\frac{u^iu^j}{2c} \left[ \int_0^L h_{ij}(t_{\rm out},\boldsymbol x_{\rm out})\,ds +\int_0^L h_{ij}(t_{\rm ret},\boldsymbol x_{\rm ret})\,ds \right].$$

Each integral is elementary for one Fourier mode. Factoring out the phase at final reception yields

$$\widetilde{\delta T}_{\boldsymbol u}(f) =\frac{L}{c}u^iu^j\, \mathcal D(f,\mu)\widetilde h_{ij}(f),$$

with the dimensionless round-trip arm transfer

$$\begin{aligned} \mathcal D(f,\mu) =\frac12\Bigl[& e^{i\pi x(3-\mu)} \operatorname{sinc}\!\bigl(\pi x(1+\mu)\bigr)\\ &+e^{i\pi x(1-\mu)} \operatorname{sinc}\!\bigl(\pi x(1-\mu)\bigr) \Bigr]. \end{aligned}$$

An overall phase changes if the response is timestamped at emission, reflection, the arm midpoint, or reception. The measurable relative phase and the magnitude are fixed once that timestamp convention is declared. The path-integral derivation and equivalent two-way timing response appear in Romano and Cornish 2017, §5.2.2, Eqs. (5.24)–(5.28), PDF.

The finite-arm Michelson tensor

For a second equal arm along $\boldsymbol v$, normalize the difference of round-trip times to strain:

$$\widetilde s(f) =\frac{c}{2L} \left( \widetilde{\delta T}_{\boldsymbol u} -\widetilde{\delta T}_{\boldsymbol v} \right) =R^{ij}(f,\boldsymbol n)\widetilde h_{ij}(f),$$

where

$$R^{ij}(f,\boldsymbol n) =\frac12\left[ u^iu^j\mathcal D(f,\boldsymbol n\cdot\boldsymbol u) -v^iv^j\mathcal D(f,\boldsymbol n\cdot\boldsymbol v) \right].$$

Projection onto a polarization tensor gives the complex antenna factor

$$F_A(f,\boldsymbol n)=R^{ij}(f,\boldsymbol n)e^A_{ij}(\boldsymbol n).$$

Frequency now changes both its magnitude and its phase. The response cannot in general be represented by a real sky pattern multiplied by a direction-independent scalar filter.

As $x\to0$, both sinc factors tend to one and all exponentials tend to one. Therefore

$$\mathcal D\to1, \qquad R^{ij}\to\frac12(u^iu^j-v^iv^j),$$

recovering the small-antenna Michelson response. This is the decisive normalization check.

Normal incidence and the relevant scale

For normal incidence, $\mu=0$, the arm transfer simplifies to

$$\mathcal D(f,0) =e^{2\pi i fL/c} \operatorname{sinc}\!\left(\frac{2\pi fL}{c}\right).$$

Its magnitude has the first null at

$$f=\frac{c}{2L}=f_{\mathrm{FSR}},$$

the inverse round-trip light time. The constant-tensor approximation is controlled by $2\pi fL/c$, not merely by whether $f$ is numerically “low.” At small argument,

$$\left\lvert\mathcal D(f,0)\right\rvert =1-\frac16\left(\frac{2\pi fL}{c}\right)^2 +O\!\left((fL/c)^4\right).$$

For kilometre-scale ground detectors, bare finite-arm suppression becomes conspicuous toward the high-frequency end of the useful band. Optical-cavity and coupled-readout poles can shape the response at much lower frequencies; those are not finite-arm effects and are treated on Fabry–Pérot Arms and Optical Cavities.

Direction, delay, and network phase

The factors $1\pm\mu$ are geometric Doppler factors. A photon and wave that nearly co-propagate accumulate phase differently from a counter-propagating pair. Apparent denominators $1\pm\mu$ in alternative forms are harmless because the accompanying numerator vanishes in the same limit; the sinc form makes regularity explicit.

A detector at position $\boldsymbol x_I$ also carries a site phase $e^{-2\pi if\boldsymbol n\cdot\boldsymbol x_I/c}$ in the convention above. That translation phase is distinct from the within-arm transfer. Ground-based networks commonly occupy the small-arm but large-separation regime: $fL/c\ll1$ for each instrument can hold while $f\lvert\boldsymbol x_I-\boldsymbol x_J\rvert/c$ is not small. Dropping both phases under one “long-wavelength” label destroys localization information.

Observational boundary and independent checks

Applying this response to data requires a detector geometry and timestamp convention, a versioned complex calibration response, a selected frequency band and data segment, and the same polarization and source-direction convention in the waveform. Public response and metadata interfaces are frozen to the GWOSC documentation available on 2026-08-18. This page does not infer a sensitivity curve or observing-state bandwidth.

Useful independent checks are:

• numerical path integration must agree with $\mathcal D$ for random $(f,\mu)$;
• $\mathcal D\to1$ as $fL/c\to0$;
• normal incidence must reduce to $e^{2\pi ifL/c}\operatorname{sinc}(2\pi fL/c)$;
• changing the timestamp may multiply the whole response by a unit phase, but cannot change its magnitude or a consistently computed inter-detector prediction.

Common pitfalls

Confusing arm length with detector separation. Finite-arm averaging and inter-site timing involve different length scales. One can be negligible while the other is essential.

Dropping the complex phase. A magnitude-only response may suffice for a power estimate but not for coherent network inference or timing.

Calling an optical pole a finite-arm null. Cavity storage, recycling, feedback, and light-path averaging are separate physical filters even when they multiply in a final response model.

Exercises

1. Derive the normal-incidence transfer. Set $\mu=0$ in the two-term expression for $\mathcal D$ and reduce it to a single sinc.

Solution

At $\mu=0$,

$$\begin{aligned} \mathcal D(f,0) &=\frac12\left(e^{3\pi ix}+e^{\pi ix}\right)\operatorname{sinc}(\pi x)\\ &=e^{2\pi ix}\cos(\pi x)\frac{\sin(\pi x)}{\pi x}\\ &=e^{2\pi ix}\frac{\sin(2\pi x)}{2\pi x} =e^{2\pi ifL/c}\operatorname{sinc}\!\left(\frac{2\pi fL}{c}\right). \end{aligned}$$

Its limit at $f=0$ is one, and its first nonzero null occurs at $f=c/(2L)$.

2. A one-percent amplitude criterion. Using the small-argument expansion, estimate the frequency at which normal-incidence finite-arm averaging reduces the amplitude by one percent for $L=4\,\mathrm{km}$.

Solution

Set $y=2\pi fL/c$ and solve $y^2/6=0.01$, giving $y\simeq0.245$. Hence

$$f\simeq\frac{0.245}{2\pi}\frac{c}{L} \simeq2.9\,\mathrm{kHz}.$$

This estimate concerns only the bare normal-incidence arm average. It does not say that a real detector has a flat calibrated response below this frequency; optical and control responses introduce other scales.

References

• Gravitational Wave Open Science Center. “API Documentation.” Version available 2026-08-18. Official documentation.
• 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.