Interferometer antenna patterns

Use an ideal equal-arm right-angle detector whose arms lie along x and y, in the long-wavelength free-mass approximation. Its tensor is one half the difference of the two arm outer products. Contracting this tensor with each transverse polarization tensor defines the corresponding antenna coefficient. The measured strain is the sum of those coefficients times the incoming polarization components, with propagation delays and optical transfer effects added when required.

Let the source line of sight have polar angle theta from positive z and azimuth phi from the x arm. The wave propagates toward the detector along minus that line of sight. At zero polarization-basis angle choose transverse axes p equal to the usual spherical theta vector and q equal to minus the spherical phi vector. This explicitly fixes the cross sign and makes the transverse axes right handed with the propagation direction.

With the chosen base axes the plus coefficient is one half of one plus cosine-squared theta times cosine twice phi, and the cross coefficient is cosine theta times sine twice phi. A passive polarization-basis rotation mixes them with the same component rotation used earlier for the strains. Their scalar contraction remains invariant when both are transformed. Changing only the source's physical polarization relative to a fixed detector is a different operation.

For an overhead source aligned with the arm basis the plus response can reach one. A source in the detector plane along an arm bisector has both coefficients zero in this ideal model. Other zeros affect only one component or a particular combination. Averaging uniformly over sky and polarization-basis angle gives mean square one fifth for each coefficient, but this does not include a binary's inclination average. Optimal, root-mean-square and population-averaged amplitudes therefore require different stated conventions.

Separated detectors sample the same wave with different antenna coefficients and geometric delays. A measured delay on one baseline constrains a direction cosine and hence a ring on the sky, rather than a unique point. Additional baselines, amplitude ratios, polarization consistency and the time evolution of detector orientation can reduce that ambiguity. Noise, calibration and finite bandwidth control how accurately those constraints can be extracted.

A single observed amplitude mixes luminosity distance with inclination, polarization and sky projection. Even a network can retain substantial degeneracies when its responses are similar or the signal is weak. The long-wavelength pattern also omits frequency-dependent optical response and arm-transfer effects. A full likelihood combines calibrated strain, noise and a propagated waveform; neither an antenna map nor an arrival-time ring alone supplies a complete astrophysical position or distance estimate.