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

TL;DR

This work tackles high-frequency noise couplings in heterodyne interferometry with phase modulation, addressing how modulation and laser phase noises couple into phasemeter readouts beyond the observation band. It develops a comprehensive analytical framework that separately treats heterodyne-band and modulation-band noises, using monotonic-noise representations, Jacobi–Anger expansions, and both trig and sideband analyses to identify coupling pathways. The authors verify the analytical results against numerical simulations and demonstrate a practical use case with LISA-like parameters to derive noise-band requirements, showing that certain noise channels can be mitigated through sideband combining while others remain intrinsic. The methods provide a systematic approach for informing phase-noise and modulator design in space-based GW detectors and other precision optical metrology systems, with implications for realistic noise budgeting and requirement setting.

Abstract

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.

Figures (8)

• Figure 1: Illustration of the frequency bands discussed in this paper. The respective frequency regimes in the LISA case are written in brackets. Throughout the paper, $\epsilon$ represents a (angular) frequency in the observation band, phase in which is to be extracted by a phasemeter. Note that noises are not shown here.
• Figure 2: Illustration of the heterodyne-band noises from \ref{['sec:het_band']}. It shows the different frequency bands and noise couplings into the various beatnotes. Note that the noises, labeled $N_\text{lsb/car/usb}$ (with their width defined by the phasemeter bandwith) and representing $a_i, p_i, \theta_i\, \&\, n_i$, are schematically shown as an additive noise floor, whilst those (except for $a_i$, as visible in \ref{['eq:spr']}) are actually direct phase noises that can be expanded in frequency by the Jacobi-Anger expressions as explained in the corresponding \ref{['sub:het_band_self', 'sub:het_band_mutual']}.
• Figure 3: Diagram comprehensively visualizing coupling of the laser and modulation noise to phase extraction by a phasemeter. Blue arrows represent the coupling of noise at the heterodyne (or observation) band $\Delta f$, discussed in \ref{['sec:het_band']}. Black arrows represent the virtual frequency conversion of noises around the integer multiple of the modulation frequency $\omega_{\mathrm{m},i}$ that are captured by the trigonometric analysis in \ref{['sub:mod_band_trig']}. Red arrows represent the modulation-band noise coupling, which can only be found in the sideband analysis in \ref{['sub:mod_band_sideband']}. The modulation-band noise coupling via the second-order modulation sidebands is not depicted. The noise scaling factors on individual paths are written in square brackets. The dashed arrows from $p_i(2f_m+\Delta f)$ to $a_i(f_m+\Delta f)$ and $\theta_i(f_m+\Delta f)$ represents the fact that the down-conversion exist but it does not couple to the heterodyne or observation band with further down conversions as derived in \ref{['eq:pi2aithetai2pi']}.
• Figure 4: Single beam optical spectrum with the laser phase noise $p_i$ in the noise sideband picture. The modulation sidebands are depicted up to the second-order. $f_{\mathrm{m},i}$ is a modulation frequency, and $\Delta f_\mathrm{p}$ is an offset frequency from an integer multiple of $f_{\mathrm{m},i}$, which can be either of heterodyne-band or observation-band frequencies. The colored noise sidebands correspond to the discussed coupling paths as follows: Red: \ref{['eq:T21_S1', 'eq:pi_2fm_viasb']}; Green: \ref{['eq:T2-2_sum']}; Blue: \ref{['eq:T12_S1', 'eq:pi_fm_viasb']}. Abbreviations: usb = upper sideband; lsb = lower sideband.
• Figure 5: Diagram highlighting the processing steps of the numerical experiment. All plots are amplitude spectra, instead of ASDs, to properly assess the height of monotonic tones. The noise coupling factor for an individual beatnote is computed by the ratio of the coupled noise in its measured phase (bottom right) to the injected noise (top left). Regarding the plots of the power spectrum, the results of the modulation amplitude noise $a_i$ at $2\omega_\mathrm{het}$ are taken as an example.