[Paper Review] Quantifying broadband chromatic drifts in Fabry-Perot resonators for exoplanet science

Molly Kate Kreider, Connor Fredrick+10 more|arXiv (Cornell University)|Oct 20, 2022

Stellar, planetary, and galactic studies4 citations

TL;DR

This paper resolves the paradox of broadband Fabry-Perot etalons exhibiting simultaneous optical length expansion and contraction by modeling chromatic drift through Fresnel analysis and transfer matrix methods. It identifies gradual relaxation of the outermost dielectric mirror layers as the dominant cause of wavelength-dependent mode drift, offering design principles for more stable etalons in high-precision exoplanet spectroscopy.

ABSTRACT

The possibility of an Earth-Sun analog beyond our solar system is one of the most longstanding questions in science. At present, answering this question embodies an extremely difficult measurement problem that requires multiple coordinated advances in astronomical telescopes, fiber optics, precision spectrographs, large format detector arrays, and advanced data processing. Taken together, addressing this challenge will require the measurement and calibration of shifts in stellar spectra at the 10^-10 level over multi-year periods. The potential for such precision has recently been advanced by the introduction of laser frequency combs (LFCs) to the field of precision astronomical spectroscopy. However, the expense, complexity and lack of full spectral coverage of LFCs has limited their widespread use and ultimate impact. To address this issue, we explore simple and robust white-light-illuminated Fabry-Perot (FP) etalons as spectral calibrators for precise radial velocity measurements. We track the frequencies of up to 13,000 etalon modes of the installed FPs from two state-of-the-art astronomical spectrographs. Combining these measurements with modeling, we trace unexpected chromatic variations of the FP modes to sub-picometer changes in the dielectric layers of the broad bandwidth FP mirrors. This yields the determination of the frequencies of the FP modes with precision approaching 10^-11/day, equivalent to a radial velocity (RV) Doppler shift of 3 mm/s/day. These results represent critical progress in precision RV measurements on two fronts: first, they make FP etalons a more powerful stand-alone calibration tool, and second, they demonstrate the capability of LFCs to extend cm/s level RV measurement precision over periods approaching a year. Together, these advances highlight a path to achieving spectroscopic calibration at levels that will be critical for finding earths like our own.

Motivation & Objective

• To resolve the paradox of an etalon appearing to expand and contract simultaneously across different spectral regions.
• To identify the physical mechanism behind wavelength-dependent mode drift in broadband Fabry-Perot etalons used for radial velocity exoplanet detection.
• To evaluate the roles of temperature gradients, incident angle variations, polarization fluctuations, and manufacturing tolerances in causing chromatic drift.
• To provide a modeling framework for predicting and mitigating mode drift in etalon-based spectrograph calibrators.
• To guide future design of broadband, stable mirror coatings for next-generation astronomical etalons.

Proposed method

• Employed the transfer matrix method to model the optical transfer function of multi-layer dielectric mirrors in the HPF etalon.
• Used Fresnel analysis to calculate the reflective phase shift of each mirror layer as a function of wavelength and incidence angle.
• Simulated mode position shifts by combining phase shifts with the round-trip phase condition for Fabry-Perot resonance.
• Systematically perturbed individual mirror layers to assess their contribution to chromatic drift, focusing on the first few layers.
• Modeled temperature gradients (linear, quadratic) and manufacturing tolerances (Gaussian thickness errors with σ = 1 nm) to assess their impact.
• Compared simulated mode drift profiles with experimental data from the HPF etalon across 800–1300 nm.

Experimental results

Research questions

• RQ1 What causes the observed paradoxical behavior where etalon modes drift in opposite directions across different spectral regions?
• RQ2 To what extent do temperature gradients, incident angle misalignment, or polarization changes contribute to chromatic mode drift?
• RQ3 How do manufacturing tolerances in mirror coating thickness affect the spectral stability of etalon modes?
• RQ4 Which mirror layer(s) are most responsible for the observed wavelength-dependent mode drift?
• RQ5 Can the relaxation of the outermost mirror coating layers explain the measured oscillatory chromatic drift?

Key findings

• The observed chromatic drift in the HPF etalon is best explained by gradual relaxation of the first one or two dielectric mirror coating layers over time.
• The relaxation process leads to a measurable decrease in refractive index and increase in thickness, consistent with the magnitude and sign of the observed mode shifts.
• Temperature gradients and incident angle variations were ruled out as primary causes due to poor qualitative and quantitative agreement with measured drift profiles.
• Manufacturing tolerances (±1 nm thickness variation) amplify low-wavelength oscillations but are insufficient to explain the full-scale drift behavior.
• The first few layers of the mirror stack produce mode shift profiles that show excellent agreement with experimental data, indicating their dominant role.
• The study provides a foundation for designing more stable broadband etalons by incorporating coating relaxation dynamics into the mirror design process.