\emph{In-situ} determination of astro-comb calibrator lines to better than 10 cm s$^{-1}$

Improved wavelength calibrators for high-resolution astrophysical spectrographs will be essential for precision radial velocity (RV) detection of Earth-like exoplanets and direct observation of cosmological deceleration. The astro-comb is a combination of an octave-spanning femtosecond laser frequency comb and a Fabry-P\'erot cavity used to achieve calibrator line spacings that can be resolved by an astrophysical spectrograph. Systematic spectral shifts associated with the cavity can be 0.1-1 MHz, corresponding to RV errors of 10-100 cm/s, due to the dispersive properties of the cavity mirrors over broad spectral widths. Although these systematic shifts are very stable, their correction is crucial to high accuracy astrophysical spectroscopy. Here, we demonstrate an \emph{in-situ} technique to determine the systematic shifts of astro-comb lines due to finite Fabry-P\'erot cavity dispersion. The technique is practical for implementation at a telescope-based spectrograph to enable wavelength calibration accuracy better than 10 cm/s.


Introduction
High precision wavelength calibrators for astrophysical spectrographs will be key components of new precision radial velocity (RV) observations, including the search for Earth-like extrasolar planets (exoplanets) [1] and direct observation of cosmological acceleration [2,3]. Recent work has demonstrated the potential of octave-spanning femtosecond laser frequency combs [4] ("astro-combs") to serve as wavelength calibrators for astrophysical spectrographs providing RV sensitivity down to 1 cm/s [5,6,7,8,9,10,11]. Exoplanet searches place stringent demands upon such calibrators. For example, the RV amplitude of the reflex motion of a solar-mass star induced by an Earth-mass planet in an orbit within the habitable zone is about 10 cm/s. The current state of the art astrophysical spectrograph, HARPS, has demonstrated stellar RV sensitivity ≈ 60 cm/s [1], largely limited by its thorium argon lamp calibrator [12,13]. These calibrators are limited by their unevenness in line spacing and intensity as well as the slow variation of their line wavelengths with time. An astro-comb provides emission lines with uniform intensity and controllable spacing, which can be referenced to atomic frequency standards and the global positioning system (GPS), yielding excellent long-term stability and reproducibility.
To date, astro-combs consist of a combination of an octave-spanning femtosecond laser frequency comb (source comb) and a Fabry-Pérot cavity (FPC), see Fig. 1. Spectral lines generated by the source comb are spaced by the pulse repetition rate ( f r ), currently ≈ 1 GHz, which results in a line spacing too dense to be resolved by broadband astrophysical spectrographs [10]. The FPC serves as a mode filter with a free spectral range (FSR) set to an integer multiple of the repetition rate, FSR=M f r , with M ≈ 10 − 100, depending on the spectrograph resolution. Ideally, the FPC passes only every M th source comb spectral line, providing thousands of calibration lines well matched to a practical spectrograph's resolution, with fractional frequency uncertainty limited only by the stability of the RF reference used to stabilize the source comb f m = f 0 +m f r Fig. 1. Block diagram of astro-comb used in measurements reported here. A 30-GHz line spacing is generated from a 1-GHz source comb mode-filtered by a Fabry-Pérot Cavity stabilized to an injected diode laser via the Pound-Drever-Hall method in transmission. The diode laser is phase locked to one source comb line, while the source comb is stabilized to an atomic frequency reference. and FPC. This frequency uncertainty can be < 10 −11 using commonly available atomic clock technology, which corresponds to ∼ 3 kHz uncertainty in the optical frequency or 0.3 cm/s in RV precision. However, because the spectrograph fails to resolve neighboring source comb lines, finite suppression of these neighboring lines by the FPC affects the lineshape and potentially the centroid of measured astro-comb lines. For example, in the results presented here, source comb modes neighboring the astro-comb line, with intensities after passing through the FPC that differ by 0.1% of the main astro-comb peak, shift the measured line centroid by 1 MHz, which corresponds to an RV systematic error of 1 m/s. In practice, such systematic RV shifts are inevitable over spectral bandwidths of 1000Å due to the dispersive properties of the mirrors of the FPC. Although these systematic shifts can be very stable over timescales of years, the correction of such shifts is crucial to high accuracy astrophysical spectroscopy.
In this paper, we demonstrate an in-situ technique to determine the systematic shifts of astrocomb lines due to FPC dispersion, which can be applied at a telescope-based spectrograph to enable wavelength calibration accuracy better than 10 cm/s. By measuring the intensity of astrocomb lines as the FPC length is adjusted, we determine (i) the offset of each FPC resonance from the nearest source comb line; (ii) FPC finesse as a function of wavelength; and (iii) the intensity of the astro-comb lines and their neighboring (normally suppressed) source comb lines. These parameters can be determined quickly and reliably over the full 1000Å bandwidth of the astro-comb with only ≈ 50 measurements at slightly different FPC lengths, and can be performed quickly (< 1 hour) and reliably. The measurement has also been performed with a lower resolution commercial optical spectrum analyzer with consistent results. The astro-comb line characterization technique presented here builds on past work in which femtosecond lasers coupled to swept cavities were used to study both the medium in which the cavity was immersed and the cavity mirrors [14,15].

Model
Imperfect suppression of unwanted source comb lines, e.g., due to finite FPC finesse, affects the astro-comb lineshape observed on a spectrograph. The lineshape can be modeled with knowl-edge of the FPC properties including mirror reflectivity and round trip phase delay. The intensity of a source comb line after the FPC is where I m is the intensity of the source comb line of optical frequency f m ; T m is the resonant transmission of the FPC at optical frequency f m ; F m is the finesse of the FPC near frequency f m ; and φ m is the round trip phase delay. The phase delay may be expressed as [17] where L is the length of the cavity, c is the speed of light in vacuum, n m is the refractive index of the medium inside the cavity (air, vacuum, etc.) at optical frequency f m , and τ( f ) is the frequency-dependent group delay of the mirrors. The first term in parentheses in Eq. (2) is the distance between the mirrors expressed in wavelengths; while the second term, the integral of τ( f ), is the phase delay of the mirrors and represents the frequency-dependent penetration distance of light into the mirror. Maximum transmission occurs when a source comb line is resonant with the FPC, or equivalently φ = 2πq, with q an integer (see Fig. 2a). The FSR, ∆, is the frequency difference between two consecutive FPC resonant frequencies. Assuming that τ varies slowly with frequency, the FSR can be approximated by For the astro-comb shown in Fig. 1, which we deployed as a wavelength calibrator for the Tillinghast Reflector Echelle astrophysical Spectrograph (TRES) [16], typical operational parameters are ∆ m ≈ 31 GHz and F m ≈ 180, with source comb lines nearest to the astro-comb peak suppressed by ≈ 22 dB (see Fig. 2b and Section 3). This imperfect suppression leads to systematic inaccuracies in astro-comb line centers at the 50 cm/s level, as observed on the TRES spectrograph across a 1000Å bandwidth. The effects of these systematic shifts can be characterized by determining the FPC finesse (F m ) and the frequency difference between astro-comb lines and the FPC resonance over the full spectral width. As described below, compensation can then be applied to determine wavelength solutions (conversions from spectrograph pixel number to wavelength) with accuracy at least an order of magnitude below the 50 cm/s level of these systematic shifts.
We determine F m and the frequency offset of the FPC resonance from the astro-comb lines by adjusting the length of the FPC and measuring the intensities of astro-comb lines over the full spectral bandwidth. When we change the frequency of one FPC resonance from f d to f d + δ f the cavity length changes by where ∆ d and τ d are the FSR of the FPC and the phase delay of the mirrors, respectively, at the frequency of the controlled resonance. The change of the cavity length leads to a change of the round trip phase delay of an astro-comb line f m by  For these settings, one source comb line out of 30 passes the FPC unattenuated, producing an astro-comb spacing of ∼ 30 GHz that can be resolved by a high-resolution astrophysical spectrograph. The wavelength of the reference diode laser is shown slightly detuned from the source comb to compensate for FPC dispersion between the diode laser wavelength and the central astro-comb wavelength as described in the text. (b) Astro-comb lines measured with the TRES spectrograph cover a bandwidth ≈ 1000Å. The gaps in the spectrum are not measured by the spectrograph due to light in these spectral regions not focusing onto its CCD. As a result, the intensity of the astro-comb line varies with the change of the frequency of one FPC mode according to Eq. (1), from which we can derive the finesse and the phase errors at all astro-comb line frequencies.

Experiment
The astro-comb employed in our experimental demonstrations is shown schematically in Fig. 1.
The octave-spanning source comb spectrum (Fig. 3) is generated by a mode-locked titaniumsapphire femtosecond laser (Octavius, Menlosystems, Inc.) with repetition rate f r ≈ 1 GHz. The absolute frequencies of the comb lines can be expressed as f m = f 0 + m f r , where m is an integer and f 0 is the carrier-envelope offset frequency. A PIN diode detects f r , which is then stabilized by adjusting the laser cavity length. The f -2 f self-referencing method is used to produce a signal at f 0 on an avalanche photodiode: comb lines around 11400Å are frequency doubled in a 1 mm thick LBO (Lithium Triborate) crystal and beat with comb lines around 5700 A. f 0 is then stabilized by intensity modulation of the 7.6 W, 5320Å pump laser. Both f r and f 0 are referenced to low-noise radio-frequency synthesizers, which are stabilized to a commercial rubidium frequency reference. For our source comb, typical values of these frequencies are: f r = 1.042048458 GHz and f 0 = 0.09 GHz, and the resulting linewidth of individual sourcecomb spectral lines is < 1 MHz. The source comb beam passes through the FPC, filtering out unwanted comb lines and increasing the transmitted line spacing. Two flat mirrors with ≈ 98.5% reflectivity and minimal group delay dispersion (< 10 fs 2 from 7500Å to 9000Å) comprise the FPC. Due to the low dispersion mirrors, the FSR of the FPC is almost constant within the 1000Å bandwidth: typically one in thirty comb lines is resonant with the FPC (Fig. 2a). The measured finesse of the FPC is ≈ 180, which is consistent with the theoretical limit estimated from the mirror reflectivity.
We stabilize the FPC by locking one transmission resonance to an injected diode laser. The diode laser is modulated at 30 MHz with an electro-optic modulator (EOM) and injected into the FPC with a polarization orthogonal to the comb light. After the FPC, the diode beam is separated from the comb, demodulated, and used to derive a feedback signal that stabilizes the FPC to the diode laser. The diode laser itself is offset-locked to one of the source comb lines at a wavelength ≈ 7950Å. The frequency of the offset lock (see Fig. 2a) is adjusted to optimize the astro-comb bandwidth, and compensates for FPC dispersion between the diode laser wavelength and the central astro-comb wavelength due to mirror properties and air in the cavity as discussed in Secs. 2 and 4. The diode laser wavelength is close to the Rb D1 transition. Absorption spectroscopy of the diode laser light using a thermal rubidium vapor cell thus identifies the diode frequency to within one source comb line. The offset lock then determines the absolute frequency of the diode laser with the accuracy of the underlying atomic frequency reference.
In its current incarnation as a wavelength calibrator for an astrophysical spectrograph, the astro-comb spectrum is analyzed by the Tillinghast Reflector Echelle Spectrograph (TRES) [16], a fiber-fed multi-order echelle spectrograph at the 1.5 m telescope of the Whipple Observatory at Mt. Hopkins in Arizona. Spectral dispersion is provided by an echelle grating with cross-dispersion from a prism operated in double-pass mode. TRES covers a spectral bandwidth from 3700Å to 9100Å with a resolving power of R ≡ λ /δ λ ≈ 50000 [16] corresponding to a resolution of ≈ 7 GHz at 8000Å. Astro-comb light is injected into an integrating sphere and then sent to TRES on a 100-µm multimode fiber. The integrating sphere reduces lineshape fluctuations due to input-dependent illumination of the spectrograph optics. Each astro-comb spectral measurement by TRES is typically integrated for 60 s and recorded on a cooled (100 K) two dimensional CCD (see Fig. 2b). Flat-field correction is applied to the astrocomb spectrum in order to remove artifacts caused by variations in the pixel-to-pixel sensitivity of the detector and by distortions in the optical path. The implementation of both the integrating sphere and the flat-field correction are essential for high accuracy astrophysical spectroscopy but is not crucial to the work described in this paper. The separation of the FPC mirrors is set such that astro-comb lines are separated by ≈ 4 resolution elements of the spectrograph. In the measurement of the intensity of each astro-comb lines, typically the counts from 18 CCD pixels around the central pixel are binned from the one dimensional extracted spectrograph spectrum to obtain each measurement points in Fig. 4. The 18 pixel integrations capture all the intensity from each astro-comb line (spectrograph resolution ≈ 6 pixels) while avoiding cross-talk between neighboring lines (line separation ≈ 24 pixels). We have also varied the binning and found no appreciable change in the result. The length of the FPC was swept (by changing the diode frequency) back and forth through the FPC resonance: we took steps of 80 MHz in one direction and then steps in the reverse direction at intermediate positions. Different step sizes or schemes have led to consistent results.
We also characterized the astro-comb spectrum with a commercial optical spectrum analyzer (OSA, ANDO AQ6315), which has much lower resolution (≈ 300 GHz) when operated in broadband (1000Å) mode. There are then ≈ 10 astro-comb lines in one resolution element. Extraction of astro-comb intensities from the OSA is significantly simpler than the two dimensional TRES spectrograph, and the OSA, therefore, provides a useful cross-check. We find that all parameters measured with the OSA are consistent with those measured with TRES. Additionally, the calibration procedure has been performed more than 10 times with both the TRES spectrograph and the OSA over 10 days. All measurements are found to be consistent over this time period.

Results and Analysis
We find good agreement between TRES measurements of the astro-comb spectrum and the model presented above. For example, Figure 4 shows the measured variation of the peak intensity of one astro-comb line vs diode laser frequency (and thus cavity length), as well as a fit of Eq. 1 to the data. From the fit, we derive the astro-comb resonant intensity T m I m ; the FPC finesse F m ; and the phase deviation δ φ m . The phase deviation is the offset of the round trip phase from an integer multiple of 2π at an astro-comb line, and is determined from the detuning of the astro-comb peak at frequency f m from center of the FPC resonance at nominal diode laser setting f FPC . The phase deviation is then given by where ∆ m is the free spectral range near frequency f m . The finesse, F, of the FPC can be derived from the fit as where ∆φ FWHM is the full width at half maximum in phase of an FPC resonance. The uncertainties in measured FPC resonance centers and widths are approximately 1 MHz, corresponding to phase uncertainties of 0.2 mrad. The variation of these parameters as a function of wavelength provides the information needed to characterize the astro-comb spectrum as a wavelength calibrator. As shown in Fig. 5, the FPC finesse and phase deviation are found to vary slowly across the entire astro-comb spectrum, as measured with TRES. The FPC finesse is approximately 180 (Fig. 5a) and the phase deviation of the FPC relative to astro-comb lines is < 20 mrad (Fig. 5b). Reflections from the back surfaces of the FPC mirrors used in these measurements produce the small, rapid phase variations observed in the figure. Despite antireflection-coatings the mirror substrates reflected 0.1% of the incident power. A simple model with realistic parameters reproduces the observed phase variations and allows systematic effects associated with these reflections to be eliminated. In future work, slight wedging (for example, 0.5 • ) of the substrates will eliminate these variations. After fitting the phase deviation with a sixth order polynomial, we compared our result with the phase deviation derived from a model of air dispersion [18] and the mirror phase delay measured with a white light interferometer (Fig. 6). The two different methods agree well, given systematic limitations to the white light interferometer measurement and resultant effect on the model fit.
Using these in-situ measurements of the FPC finesse and the phase deviation, along with the source comb intensity variation and resonant FPC transmission, we can determine the suppression of unwanted source comb lines and the resultant frequency shifts of astro-comb line centers. The variation with optical frequency of source comb line intensity and resonant FPC transmission (T m · I m in Eq. 1) is measured by offsetting the diode laser frequency from its nominal value by the source comb repetition rate, and thereby tuning the FPC to resonance with a neighboring (and normally suppressed) source comb mode. The maximal transmission of each transmitted source comb line and thus T m · I m is measured; and then the procedure is repeated for all normally suppressed source comb modes. We find that the variation of T m · I m is within our measurement uncertainties (< 1%). Evaluating Eq. 1 for a single astro-comb line and all of its neighboring (suppressed) source comb lines within one spectrograph resolution element, and determining the net transmitted spectrum weighted by the frequency offset from the central astro-comb frequency, allows us to evaluate the effective line center shift in the astro-comb spectrum. For moderate finesse and f r ≫ ∆ m /F, the systematic shift δ f m of an astro-comb line of frequency f m , as measured on a spectrograph, can be approximated as where the term in parentheses parameterizes the mean suppression of nearest source comb side modes, the first term in square brackets results from the difference in source comb intensity between the upper and lower side modes, and the second term in square brackets results from FPC phase deviation leading to asymmetry of the comb lines relative to the FPC mode. The frequency shift in an astro-comb line centroid due to variations of neighboring source comb line intensities at the 1% level is thus ≈ 70 kHz or 7 cm/s. Note that we do not expect neigh- boring source comb lines to differ as much as 1%; also a Gaussian distribution of comb-line intensities will average away much of this astro-comb centroid frequency shift. Nonetheless, we expect that a double pass configuration through the same FPC [19,20], moderately higher FPC finesse (F ≈ 500), a higher repetition rate for the source comb ( f r ≈ 5 GHz), or more precise measurements of line to line intensity variations will be required to assure 1 cm/s accuracy In practical astro-comb calibrators. In Fig. 7, we show the shift of the estimated systematic error in the wavelength calibration of TRES as a function of wavelength, when using our current astro-comb as the calibration reference, if the effect of nearest suppressed source-comb lines are not included in the fit model. With proper inclusion of such effects, residual uncertainty in the wavelength calibration is at the 1 cm/s level. Thus we conclude that while astro-comb line frequency shifts caused by mirror dispersion and resultant wavelength dependence of FPC finesse and phase deviation can be absorbed into the wavelength calibration if reproducibility is all that is desired, at the few cm/s level these corrections must be applied to the spectrograph calibration to achieve accurate stellar radial velocity measurements.

Conclusion
In summary, we have demonstrated an in-situ method to determine systematic shifts of astrocomb spectral lines due to imperfect suppression of source comb lines by the Fabry-Perót filter cavity (FPC). This method involves measurement of FPC finesse as well as the phase deviation of all astro-comb lines over a bandwidth of 1000Å. Such measurements can be performed at a telescope with either an astrophysical spectrograph or a commercial optical spectrum analyzer of lower resolution. From the measured phase deviation, the dispersion of the FPC is derived and found to be consistent with that calculated from the intra-cavity air dispersion and the group delay dispersion of the FPC mirrors. Applying resultant corrections for shifts in the centroid of astro-comb spectral lines allows us to model the astro-comb spectrum as measured on an astrophysical spectrograph with accuracy to better than 0.1 MHz, i.e., 10 cm/s for measurement of a stellar radial velocity. Such high accuracy is important for many applications of astro-comb wavelength calibrators, such as the search for habitable exoplanets, direct measurement of the expansion of the universe, and searches for a temporal variation of physical constants.