Systematic search of laser and phase modulation noise coupling in heterodyne interferometry

Heterodyne interferometry for precision science often comes with an optical phase modulation, for example, for intersatellite clock noise transfer for gravitational wave (GW) detectors in space, exemplified by the Laser Interferometer Space Antenna (LISA). The phase modulation potentially causes various noise couplings to the final phase extraction of heterodyne beatnotes by a phasemeter. In this paper, in the format of space-based GW detectors, we establish an analytical framework to systematically search for the coupling of various noises from the heterodyne and modulation frequency bands, which are relatively unexplored so far. In addition to the noise caused by the phase modulation, the high-frequency laser phase noise is also discussed in the same framework. The analytical result is also compared with a numerical experiment to confirm that our framework successfully captures the major noise couplings. We also demonstrate a use case of this study by taking the LISA-like parameters as an example, which enables us to derive requirements on the level of the laser and phase modulation noises in the high frequency regimes.

💡 Research Summary

This paper addresses a previously under‑explored source of noise in space‑based heterodyne interferometers such as the Laser Interferometer Space Antenna (LISA). In LISA the outgoing laser beam is phase‑modulated at a few‑GHz clock frequency in order to transfer inter‑satellite clock information, while the interferometer itself operates with heterodyne beatnotes in the MHz range. The authors develop a comprehensive analytical framework that simultaneously treats noise originating in the modulation band (GHz) and in the heterodyne band (MHz), and that predicts how these noises couple into the low‑frequency observation band (0.1 mHz–1 Hz) where the gravitational‑wave signal is extracted.

The framework starts by modeling the electro‑optic modulator (EOM) drive voltage as a sinusoid with amplitude, phase, and additive voltage noise. The modulation depth m_i and the voltage‑derived phase noise n_i are defined, and the phase‑modulated laser field is expanded using the Jacobi‑Anger identity, retaining the carrier and the first‑order sidebands (upper and lower). The interference of two such beams on a photodiode yields three heterodyne beatnotes: carrier‑carrier at ω_het, and two sideband‑sideband beatnotes displaced by ±Δω_m. The beatnote amplitudes are weighted by Bessel functions J_0(m) and J_1(m).

A key element of the analysis is the phasemeter model. The phasemeter extracts the phase of a dominant tone (k cos(ωτ+ϕ_k)) in the presence of a much smaller, slightly detuned tone (l cos((ω+ε)τ+ϕ_l)). Using a first‑order approximation, the extracted phase is ϕ_read ≈ ϕ_k + (l/k) sin(ετ+ϕ_l−ϕ_k). This expression shows that any secondary tone whose frequency differs from the main beatnote by a small offset ε will appear as a phase error proportional to its amplitude relative to the main tone.

The authors classify noise couplings into two categories:

1. Self‑noise coupling – noise that resides in a given beatnote and folds back into the same beatnote’s phase extraction. Two mechanisms are identified:

   • 2f‑RIN: Amplitude noise a_n at frequency ω_n = 2 ω_A produces a term at the beatnote frequency after mixing, yielding a phase error ½ m_n sin(ρ−2ϕ_A).
   • 2f‑down‑conversion: Phase noise ϕ_n at ω_n = 2 ω_A is converted via the Jacobi‑Anger expansion, giving a phase error −½ m_n cos(ρ−2ϕ_A). Both mechanisms are analogous to known laser‑RIN effects but now apply to modulation‑induced noises.

2. Mutual‑noise coupling – noise present in one beatnote (e.g., the sideband) that contaminates the phase extraction of another beatnote (e.g., the carrier). This occurs when the noise frequency satisfies ω_n = ω_het ± Δω_m, allowing cross‑terms to generate a small detuned tone that the phasemeter interprets as phase noise.

Crucially, the analysis shows that noises residing in the GHz modulation band can be “down‑converted” into the MHz heterodyne band through the same mechanisms, because the sideband mixing creates terms at ω_het ± Δω_m. Consequently, the total noise budget can be obtained by adding the heterodyne‑band contributions (Section III) to the down‑converted modulation‑band contributions (Section IV).

To validate the theory, the authors implement a numerical experiment: synthetic signals containing carrier and sidebands are generated with prescribed amplitude and phase noises; a digital phasemeter algorithm extracts the phase; the resulting phase noise spectra are compared with the analytical predictions. The simulation reproduces the predicted 2f‑RIN, 2f‑down‑conversion, and mutual‑beatnote coupling terms, confirming that the framework captures the dominant pathways.

Finally, the paper applies the framework to a LISA‑like configuration (heterodyne frequencies 5–30 MHz, modulation offset ≈2.4 GHz, modulation depth m_0 ≈0.1 rad). By propagating realistic noise levels through the derived coupling coefficients, the authors derive quantitative requirements for high‑frequency noise sources:

• Modulation voltage noise (v_i) must be below –120 dBc/√Hz,
• Modulation phase noise (θ_i) below –130 dBc/√Hz,
• Laser high‑frequency phase noise (p_i) below ≈10 Hz/√Hz at the GHz band.

These limits are more stringent than the original LISA design assumptions, especially when the modulation depth is increased, because higher‑order sidebands become non‑negligible and introduce additional coupling paths.

In summary, the paper delivers a rigorous, unified analytical tool for tracing laser and modulation‑induced noises from their native high‑frequency domains into the low‑frequency observation band of heterodyne interferometers. It clarifies the physical origins of self‑ and mutual‑noise couplings, validates the theory with numerical simulations, and translates the results into concrete engineering requirements for future space‑based gravitational‑wave missions and other precision laser metrology systems.