Quantifying broadband chromatic drifts in Fabry-Pérot resonators for exoplanet science Molly Kate Kreider1 2 3 Connor Fredrick1 2 3Scott A. Diddams1 2 3 Ryan C.

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Quantifying broadband chromatic drifts in

Fabry-Pérot resonators for exoplanet science

Molly Kate Kreider,1, 2, 3, ∗Connor Fredrick,1, 2, 3 Scott A. Diddams,1, 2, 3, †Ryan C.

Terrien,4Suvrath Mahadevan,5, 6 Joe P. Ninan,7Chad F. Bender,8Daniel Mitchell,9

Jayadev Rajagopal,10 Arpita Roy,11 Christian Schwab,12 and Jason T. Wright5, 6, 13

1Department of Physics, University of Colorado Boulder, 440 UCB Boulder, CO 80309, USA

2Electrical Computer and Energy Engineering, University of Colorado Boulder, 425 UCB Boulder, CO 80309, USA

3Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80305, USA

4Department of Physics and Astronomy, Carleton College,

One North College Street, Northﬁeld, MN 55057, USA

5Department of Astronomy & Astrophysics, 525 Davey Laboratory,

The Pennsylvania State University, University Park, PA 16802, USA

6Center for Exoplanets and Habitable Worlds, 525 Davey Laboratory,

The Pennsylvania State University, University Park, PA 16802, USA

7Dept. of Astronomy and Astrophysics, Tata Institute of Fundamental Research,

1 Homi Bhabha Road, Colaba, Mumbai -400005, India

8Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ 85721, USA

9LightMachinery, Inc., 80 Colonnade Rd N, Unit 1, Nepean, ON K2E 7L2, CA

10NSF NOIRLab, 950 N Cherry Ave, Tucson, AZ 85179, USA

11Schmidt Sciences, New York, NY 10011, USA

12School of Mathematical and Physical Sciences, Macquarie University,

Balaclava Road, North Ryde, NSW 2109, Australia

13Penn State Extraterrestrial Intelligence Center,

525 Davey Laboratory, Penn State, University Park, PA, 16802, USA

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 measure-

ment problem that requires multiple coordinated advances in astronomical telescopes, ﬁber optics,

precision spectrographs, large format detector arrays, and advanced data processing. Taken to-

gether, 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 astro-

nomical 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-Pérot (FP) etalons as spectral calibrators for precise radial ve-

locity 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.

I. INTRODUCTION

The discovery and characterization of an Earth-mass

planet orbiting a Sun-like star at 1 AU (a.k.a. Earth

2.0) is a long standing challenge in astrophysics–with pro-

found implications regarding the uniqueness of Earth, the

formation of planetary systems, and the conditions un-

der which life could exist elsewhere in the Universe. As a

result, tremendous eﬀort has been focused on this prob-

∗mollykate.kreider@colorado.edu

†scott.diddams@colorado.edu

lem, with the dominant techniques of planetary transits

and radial velocity measurements being used in tandem.

While a transit, or the passage across the stellar disk,

provides information on the size of an exoplanet, mea-

surements of the stellar radial velocity (RV) provide in-

formation on the exoplanet mass through the periodic

Doppler shifts of the stellar spectrum (Fig. 1a, b). How-

ever, RV detection of an Earth analog requires Doppler

precision of a few cm/s (fractionally 10−10) which must

be maintained over time scales of multiple years. Ul-

timately, such detection could be followed by spectro-

scopic measurements made by instruments like the James

Webb Space Telescope to identify biomarkers in the at-

mosphere.

arXiv:2210.10988v2 [physics.optics] 28 Apr 2024

Telescope

Star

Exo-

planet

Telescope

Star Exo-

planet

(a) (b)

Stellar Spectrum

Calibration Source

①②③

①

②

③

(c)

(a)

(b)

Laser Frequency Comb

Vacuum

ULE

Fabry-Perot Etalon

Echelle Spectrograph

time

position on detector array

\\ \\

!" #!

"$ !

%$" & '()

(* + ,"

time

fractional frequency shift (-!.!/

slope = ∆(&'

⁄

∆*

0!1 &-!

∆(#$

⁄

∆'

wavelength

mode shift ("#$%

+& %

'"()

○ : frequency comb

●: FP etalon

Supercontinuum

Source

Mode-locked Laser or

Electro-optic Comb

GPS

(d)

(e) (f) (g)

Earth

2.0

FP Etalons & Laser

Frequency

Combs

Atomic Lamps

Measurement Time (min)

Calibration Precision (cm/s)

Figure 1. (a) Principle of radial velocity exoplanet detection measurement, which uses (b) a known calibration source to

measure the periodic Doppler shift of absorption features in a stellar spectrum due to the gravitational pull of an exoplanet. (c)

Calibration precision levels needed to detect an Earth-Sun analog, as well as demonstrated precision levels for several diﬀerent

classes of calibrators. (d) Experimental setup of chromatic drift measurements on the HPF and NEID etalons, as detailed in

[1]. An echelle spectrograph is alternatingly illuminated by laser frequency comb and etalon light. (e) The spatial location (i.e.,

frequency) of each mode drifts over time, as illustrated in for three modes. (f) The drift of the frequency comb is indicative of

instrument drift, and the change in spacing between a frequency comb – etalon mode pair, ∆(δfn), is indicative of the etalon

drift. The fractional frequency shift for any given mode is linear in time. (g) The slopes of the fractional frequency shift can

be plotted across the wavelength range of interest, showing the complicated chromatic drift of the etalon [1].

This is an extremely challenging and multifaceted mea-

surement that requires simultaneous advances along mul-

tiple fronts [2]. Presently, some of the most outstanding

challenges are due to stellar surface activity and variable

telluric contamination from our own atmosphere, both

of which look like “noise” and mask the desired center-of-

mass RV Doppler shift, as well as increase the time and

eﬀort it takes to realize long-term measurement cadences

with suﬃcient photon signal-to-noise ratio. Equally im-

portantly are the stable spectrographs and the precise

spectrograph calibration that need to provide the sta-

ble reference against which tiny Doppler shifts would be

measured. Conventional sources for calibration include

atomic emission lamps and gas absorption cells, both of

which have intrinsic limitations in uniformity and un-

known long term stability that hinder their precision [3].

Laser frequency combs (LFCs) provide a broad spec-

tral array of discrete and evenly-spaced emission lines

that can be fully stabilized to absolute frequency stan-

dards [4]. These properties make LFCs a near ideal spec-

trograph calibration source [5, 6], and in recent years,

they have shown fractional calibration precision below

10−10 [7, 8]. However, they lack full spectral coverage in

the blue. Moreover, their implementation remains costly

and complex, and is therefore currently limited to a few

RV instruments situated at premier facilities. This mo-

tivates the use of an alternative calibrator. Moreover, no

calibration technique (frequency combs included) has yet

demonstrated the capability to resolve 1 cm/s Doppler

shifts over timescales of years or longer. As summa-

rized in Fig. 1c, there is a gap between the experi-

mentally demonstrated timescales over which etalon- and

LFC-calibrated spectrographs have achieved this preci-

sion level and the timescales which will be necessary to

detect an Earth-Sun analog.

Our work addresses this issue by exploring new capa-

bilities and resolving systematic uncertainties in Fabry-

Pérot etalons, which have long been used in astronomi-

cal spectroscopy [9]. Broadband, white-light illuminated

etalons continue to play an important role in astronomy

as a simple, robust calibration source in radial velocity

(RV) exoplanet detection. They are used as primary cal-

ibrators (in combination with lamps) in some spectro-

graphs (e.g. [10]), and in others, they extend the overall

spectral coverage to blue wavelengths not easily achieved

with LFCs, but they lack long-term absolute stability.

We investigate this shortcoming through the analysis

and modeling of two extended 6-month cross-calibration

studies that resolve puzzling chromatic drifts that have

appeared in multiple installed FPs [1]. Speciﬁcally, we

show that, with the help of an LFC, it is possible to track

and characterize ∼13,000 modes of a FP with precision

that translates to sub-picometer resolution of the optical

length of the FP. When combined with careful modeling

of relaxation in the FPs dielectric coating, we can dis-

entangle drifts consistent with an achromatic change in

cavity length versus higher-order drifts with equivalent

RV residuals at a level near 3 mm/s (10−11). This new

understanding of the complicated drifts of broad band-

width FP cavities informs future mirror design parame-

ters and immediately improves the achievable precision

level when using them as stand alone calibrators that are

periodically referenced to absolute frequency standards.

II. MEASURING CHROMATIC MODE DRIFT

IN FP ETALONS WITH LFCS

Within the landscape of calibration sources, FP etalons

are particularly attractive due to their unique combina-

tion of high spectral resolution and information content,

similar to that of an LFC [11], but with relative simplic-

ity and cost eﬃciency. They have become increasingly

commonplace in high-precision radial velocity spectro-

graphs, and are used as calibrators in a number of major

instruments at large telescopes, including CARMENES

[12], Espresso [13], HARPS [14], HPF [3], KPF [15],

MAROON-X [16], NEID [17], NIRPS [18], and SPIRou

[19]. They are also slated to be implemented with new

instruments to be installed at large telescopes, including

iLocator [20] and G-CLEF [21].

While mode-locking ﬁxes the line spacing of an LFC,

the modes of an FP cavity are determined by the local

resonance condition, which is subject to the wavelength-

dependent optical path length of the cavity geometry and

mirror coatings. The stability of its spectra is tied to

the stability of the cavity’s mechanical and optical prop-

erties, which can drift with time. Ultra-stable etalons

that exhibit a minimal amount of drift have been devel-

oped for precision spectroscopy and optical clocks [22].

Single-frequency near-infrared lasers locked to the modes

of silicon resonators held at cryogenic temperatures at a

zero crossing of the coeﬃcient of thermal expansion have

demonstrated drift rates below one part in 10−13/day

[23–25]. While not as performant as those designed for

cryogenic temperatures, cavities built using room tem-

perature ultra-low expansion (ULE) materials operate in

the visible region of the spectrum. Such cavities are still

capable of achieving fractional stabilities at 10−11 from

day to day (equivalent to a frequency drift of ∼0.1Hz/s

at 1µm) [26–28], and are particularly attractive for as-

tronomical spectroscopy.

One method of circumventing limitations due to drift

is to continuously reference one mode of an etalon to an

absolute frequency standard, like an atomic transition,

and then extrapolate the behavior of that single mode

to the behavior of the etalon’s entire spectrum [29–31].

However, such a technique is only capable of correcting

drift in an etalon to the level at which the drift behavior

is achromatic. Such a model assumes a fractional change

of cavity length ∆L/L is equal to the fractional frequency

change ∆f/fof any cavity mode. This model is applica-

ble when the gradual relaxation of etalon spacer sets the

cavity length [32], and it is well-established that ULE

cavity spacers do indeed drift over time [33–35].

However, measurements made at several etalon sys-

tems employed in precision radial velocity instruments

have shown that etalon mode drift can be signiﬁcantly

and unexpectedly more complex [1, 13, 36]. Utilizing

LFCs, we have made comprehensive and long-term mea-

surements of the etalons used with the Habitable-zone

Planet Finder (HPF) and NEID spectrographs. The

unique, broadband nature of these studies, which mea-

sure the average daily drift rate of over approximately

5,000 modes (or hundreds of nanometers) of the two FP

etalons, is made possible by leveraging the instrument

architecture necessary for high precision radial velocity

detection.

A simpliﬁed schematic of the etalon characterization

is illustrated in Figure 1d. An echelle spectrograph

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QuantifyingbroadbandchromaticdriftsinFabry-PérotresonatorsforexoplanetscienceMollyKateKreider,1,2,3,∗ConnorFredrick,1,2,3ScottA.Diddams,1,2,3,†RyanC.Terrien,4SuvrathMahadevan,5,6JoeP.Ninan,7ChadF.Bender,8DanielMitchell,9JayadevRajagopal,10ArpitaRoy,11ChristianSchwab,12andJasonT.Wright5,6,131Departme...

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Quantifying broadband chromatic drifts in Fabry-Pérot resonators for exoplanet science Molly Kate Kreider1 2 3 Connor Fredrick1 2 3Scott A. Diddams1 2 3 Ryan C.

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