Single-mode waveguides for GRAVITY II. Single-mode fibers and Fiber Control Unit

The second generation Very Large Telescope Interferometer (VLTI) instrument GRAVITY is a two-field infrared interferometer operating in the K band between 1.97 and 2 . 43 µ m with either the four 8m or the four 1.8m telescopes of the Very Large Telescope (VLT). Beams collected by the telescopes are corrected with adaptive optics systems and the fringes are stabilized with a fringe-tracking system. A metrology system allows the measurement of internal path lengths in order to achieve high-accuracy astrometry. High sensitivity and high interferometric accuracy are achieved thanks to (i) correction of the turbulent phase, (ii) the use of low-noise detectors, and (iii) the optimization of photometric and coherence throughput. Beam combination and most of the beam transport are performed with single-mode waveguides in vacuum and at low temperature. In this paper, we present the functions and performance achieved with weakly birefringent standard single-mode fiber systems in GRAVITY. Fibered di ff erential delay lines (FDDLs) are used to dynamically compensate for up to 6mm of delay between the science and reference targets. Fibered polarization rotators allow us to align polarizations in the instrument and make the single-mode beam combiner close to polarization neutral. The single-mode fiber system exhibits very low birefringence (less than 23 ◦ ), very low attenuation (3.6-7dB / km across the K band), and optimized di ff erential dispersion (less than 2 . 04 µ radcm 2 at zero extension of the FDDLs). As a consequence, the typical fringe contrast losses due to the single-mode fibers are 6% to 10% in the lowest-resolution mode and 5% in the medium-and high-resolution modes of the instrument for a photometric throughput of the fiber chain of the order of 90%. There is no equivalent of this fiber system to route and modally filter beams with delay and polarization control in any other K-band beamcombiner.


Introduction
Telescopes (ATs).Adaptive-optics systems correct for turbu-7 lence in the individual pupils and fringes are stabilized with 8 a fringe-tracking system.When not the science target itself, a 9 reference source for fringe tracking can be selected in a small 10 field around the science target (2 arcsec with the UTs, 6 arc-11 sec with the ATs) in order to perform double-field interferom-12 etry.A metrology system allows the two fringe systems to be 13 connected in order to perform high-precision narrow-angle as-14 trometry or phase-referenced imaging.Images can also be re-15 constructed with GRAVITY using closure phases.The instru-16 ment is presented in GRAVITY Collaboration et al. (2017).The 17 present paper outlines the rationale for using single-mode fibers 18 in GRAVITY and the characteristics of the fibers used in the instrument as well as of the functions performed by the fibers; it is the second in a series of papers on the single-mode waveguides of GRAVITY and follows Perraut et al. (2018).
The use of single-mode fibers for astronomical interferometry was first suggested by Froehly (1981).First tests in the laboratory (Shaklan & Roddier 1987) or in an astronomical environment (Connes & Reynaud 1988) showed the potential of the technique.The use of single-mode fibers was first motivated by their potential ability to clean the beams from the aberrations induced by atmospheric turbulence, which were the cause of poorly calibrated visibility measurements.The use of singlemode fibers for interferometry was established with the Fiber Linked Unit for Optical Recombination (FLUOR) instrument at Kitt Peak with the McMath telescope, where the ability to detect astronomical fringes was demonstrated (Coudé du Foresto & Ridgway 1992).FLUOR was upgraded and moved to the Infrared-Optical Telescope Array (IOTA) interferometer at the Fred Lawrence Whipple observatory where the accurate calibra-A&A proofs: manuscript no.47587corr_GP_v2_(without_red_text) Fig. 1.Overview of the GRAVITY chain of single-mode waveguides.The light coming from the telescopes is injected in the Fiber Coupler fibers inside the cryostat of GRAVTY (first section).Each fiber is then connected to a FDDL (section 2) and to a Fibered Polarization Rotator (section 3).The light is then fed to the V-groove fibers connected to the integrated optics beam combiner (section 4 inside the dash-dotted line box).All work at cold temperatures (240 K for the first three sections and 200 K for the integrated optics beam combiner part).This figure was first presented in Perraut et al. (2018).
tion of visibilities was effectively demonstrated, leading to the entation of polarizations needs to be adjusted to maximize fringe contrast (Section 5).
Given the ambition of GRAVITY to perform very highaccuracy astrometry on a source more than 5 magnitudes fainter than the then faintest source ever observed in optical interferometry, we made the choice to use standard (i.e., nonpolarizationmaintaining) single-mode fluoride glass fibers feeding an integrated optics beam combiner.The fluoride-glass fibers indeed have high throughput and we could contemplate a throughput as high as 90% for the whole chain of single-mode fibers of Fig. 1.We had built the know-how for high throughputs with singlemode fibers with the 'OHANA project and we knew the transmission to expect for the approximately 20 meters of fiber that we thought would be necessary for GRAVITY.As there is no need to split light after standard fibers, this choice would also lead to high throughputs in particular when there is no scientific need to split polarizations to measure the Stokes parameters, thereby saving some precious percentage of throughput while the alignment of polarizations can be achieved in the fibers with fiber polarization rotators (see Section 5).We also knew that fluorideglass fibers could potentially have very low birefringence, meaning two polarizations could be easily mixed without significant loss of contrast and that the weak birefringence would not prevent high-accuracy astrometry.In addition, delays can also be built inside the fibers by stretching the waveguides (Section 4), allowing us to balance optical paths in double-field interferometry inside the fibers.At the end of the chain, the fibers feed the single-mode, integrated optics beamcombiner (Perraut et al. 2018).As a consequence, all the functions required to achieve the specifications of GRAVITY could a priori be made with standard fibers while ensuring higher throughput and flexibility than with any alternative solution.This choice has been validated by the performance achieved by GRAVITY, the most sensitive interferometric instrument ever built with very high accuracy capabilities (GRAVITY Collaboration et al. 2017).
Alternative concepts of fiber interferometers exist for the K band.Low-OH fibers are used for the Michigan Young STar Imager (MYSTIC) instead of fluoride glass fibers (Setterholm et al. 2023) but with a much higher attenuation: 269 dB/km compared to 4.71 dB/km at 2.2 µm.This therefore forces MYSTIC fixed point in the field but cannot compensate for path-length differences between the two targets of GRAVITY.The purpose of the FDDLs is to compensate for the residual differences, which can be as large as 6 mm for two targets at low elevation separated by 6 arcsec and with a differential velocity of up to 150 nm/s with the 200 m maximum baseline of the VLTI using the ATs.We note that this stroke is not enough for the GRAVITY wide mode of GRAVITY+, and an additional system is required to compensate differential delays in this mode (GRAVITY+ Collaboration et al. 2022).Another purpose of the FDDLs is to offset the bias in optical path difference (OPD) between the fibers after minimizing differential dispersion (see Sections 3.3 and 4).In addition, the FDDLs need to be stable enough to get negligible OPD noise during long integrations compared to other sources, with the goal being to reach 1 nm over 100 s exposures.
Because of the use of weakly birefringent standard fibers, the differential angles between the polarizations of the GRAVITY beams need to be zeroed before injection into the integrated optics chips.The requirement on the FPRs is to align the polarization axes with an accuracy of better than 8 • in order to reach contrasts of 99% or higher.Although they are motorized, the FPRs are not used to dynamically compensate for polarization rotations in both GRAVITY and VLTI but to make GRAVITY polarization neutral.In addition, the user has the option to split and analyze polarizations with a half-wave plate downstream from the beam combiners in the spectrometers in order to measure the Stokes parameters and perform interfero-polarimetry.
Three series of four FPRs and four FDDLs have been produced and labelled ABCD, EFGH, and IJKL.The last batch is a spare, while the first two are used in the GRAVITY instrument.ABCD are used in the Fringe Tracker channels, while EFGH are used in the Science channels.The IJKL spares have been individually characterized, but the performance (differential dispersion and throughput) of the full chain with the IJKL FDDLs and FPRs cannot be given as they have not been connected to the Fiber Coupler and V-groove fibers.
The performance of the fibers and FCU and the specific performance of the FDDLs and FPRs are given in Table 1.Some characteristics of the fibers and of the FCU were already presented in GRAVITY Collaboration et al. (2017) and Perraut et al. (2018).These are updated and discussed in more detail in the following sections.

Characteristics of the fibers
The fibers for GRAVITY were developed by the Le Verre Fluoré company in France.The class of ZBLAN glasses was discovered by Poulain et al. (1975) with promising transmission properties in the infrared.The fibers used for GRAVITY are a type of ZBLAN mostly made of zirconium (ZrF4).
The main function of the fibers is to transport light across the instrument with minimum losses.For single-mode interferometry, another function is to filter the beams to get rid of the variations of phase due to turbulence across individual pupils.Doing so, the phase fluctuations are traded against intensity fluctuations, which are measured and calibrated out following the principles introduced for the FLUOR instrument (Coudé du Foresto et al. 1997).In the case of GRAVITY, the so-called photometric signals -which measure the quantity of photons coupled in each fiber-are extracted from the 24 outputs of the beam combiner by linear combination.
Apart from the light pollution issue experienced with the metrology laser (Section 3.5), no major difficulty was experienced when using the fibers at 240 K, even when stretching and length λ; the radius of the cladding is supposed to be infinite.

252
The fiber is single-mode for wavelengths λ larger than the cutoff wavelength λ c = 2πaNA 2.405 (Neumann 1988) Gaussian if the index law is Gaussian across the fiber) but is a 269 combination of Bessel functions.However, it is relatively close to a Gaussian beam and a waist is defined, the value of which is approximately: w 0 a 0.65 + 1.619 V 3/2 + 2.879 V 6 , with V = 2πaNA λ being the normalized frequency (Neumann 1988).The f /D ratio of the focusing optics is optimized to reach optimum injection in the fiber in the absence of aberrations.The maximum coupling efficiency for a uniform and full circular pupil is 78% (Shaklan & Roddier 1988).The waist varies almost linearly with the wavelength, which means that the injection efficiency is almost flat across the K band.NA and a are chosen so that an aberrationfree focusing optics can be easily produced.Together with the constraints to be fulfilled to produce fibers of excellent quality, this led to the choice of NA = 0.22 and 2a = 6.55 µm at 2 µm (meaning f /D 3 for the focusing optics) as shown in Table 1.With these values, the cut-off wavelength reaches 1.88 and 1.9 µm for the two batches of fibers used for GRAVITY.This was confirmed by measuring the fiber throughput for increasing curvature radius, which set the cut-off wavelength at 1.9 µm.

Birefringence
The GRAVITY fibers were specified to be at most weakly birefringent in order to allow operation without splitting polarizations while reaching high fringe contrasts.The aim here is to achieve maximum sensitivity when working with faint sources and to perform astrometry between two sources with optimum accuracy.The effect of birefringence between the two axes of polarization can be characterized by measuring the generalized Malus law of the delay lines and the FPRs of Section 5 with the setup explained in Appendix A. The intensity for various positions of the polarizer and of the analyzer between 0 and Table 2. Birefringence measurements of the GRAVITY fibers at 240 K.The test samples are cables made of a fiber from the Fiber Coupler, a FDDL, and a FPR.Roughly 2 m of fibers at ≈ 290 K were connected to the polarization bench outside the cool vessel.The FDDLs and FPRs are paired (labels A to L) as used on the instrument and labeled accordingly as either Fringe Tracker or Science Channel.Telescope is the telescope number in the GRAVITY instrument.Telescope 1 is fed by either AT4 or UT4, Telescope 2 by AT3 or UT3, etc.The basic measurement is the differential birefringence phase δϕ.No error bars are given but the method allows to detect phases as low as 0.05 rad, giving an idea of the accuracy of the measurements.The beating length L B and the minimum and maximum contrasts are deduced from this quantity.The ranges of fringe contrasts are obtained by pairing a fiber with the fibers of closest or most different measured birefringence in a given channel, the lowest contrast being obtained by adding the birefringent phases, and the highest contrasts by subtracting them.
Channel Telescope Beam Fiber length (m) δϕ (rad) L B (m) Min-max contrast (%)  305 where θ P is the angle of the polarizer, θ A is the angle of the an-306 alyzer, and δϕ is the amount of differential birefringence phase 307 between the two axes of polarization.The effect of birefringence 308 is to reduce the amplitude depending on the angle of the polarizer 309 in the input.There is no effect for a polarization aligned with the neutral axes of the fibers, that is, for θ P = π 2 ( π 2 ), and the effect is maximum for a polarization rotated by 45 • (90 • ), that is, for . The envelope of the family of generalized Malus laws is no longer flat, as shown in Fig. B1 of Appendix B. This effect shows up in Fig. 2 where the envelope is plotted with the black dashed lines.This figure is compatible with the existence of neutral axes for the fiber, which in this case are aligned with the 0 and 90 • positions of the polarizer.Equation B.4 of Appendix B gives the expressions of the upper and lower envelopes: . (2) These expressions depend on the single parameter δϕ and provide a method to measure the effect of birefringence.In the presence of birefringence, the maximum of the minimum is no longer zero and the minimum of the maximum is no longer 1, but these are respectively sin 2 δϕ 2 and 1 − sin 2 δϕ 2 , assuming that |δϕ| ≤ π 2 , that is, for a length of fiber equal to less than onequarter of the birefringence length with a maximum modulation of the Malus laws in this particular case.The difference between the propagation constants of the two polarization axes can be written as: where L B is the beating length of birefringence.The accumulated differential phases between the two polarizations over a length of fiber L can be written as: and therefore for a given fiber length L, the beating length of birefringence can be deduced from the measurement of δϕ, as: The measurement of the birefringence characteristics of the fibers at 240 K using the method presented in this paper is given in Table 2.The beat length of birefringence was measured between 297 m and 1110 m for the GRAVITY fibers.
Fringe contrasts of interferograms are degraded by the difference in differential birefringence phase ∆δϕ between the beams as they are multiplied by cos ∆δϕ 2 (see e.g., Rousselet-Perraut et al. (1996)).As the orientations of the neutral axes of the fibers have not been controlled in GRAVITY, the worst case for the fringe contrast is obtained when the differential phases are large, have opposite signs, and add up in the above formula as  been applied for GRAVITY.The dispersion of each element of 398 GRAVITY (fiber couplers, FDDLs, FPRs, and fiber bundles to 399 the integrated optics chips) has been measured against a com-400 mon reference in a Mach-Zehnder interferometer at Le Verre 401 Fluoré, which was developed with LESIA in the 1990s.A fused-402 fluoride glass-fiber coupler was used as a beam combiner for 403 accurate and repeatable dispersion measurements.The various 404 fiber components of GRAVITY were then assembled to mini-405 mize both path length and dispersion differences, which reached 406 a maximum of 2.17 mm in pathlength (B-C baseline of the FT 407 channel) and 2.04 µrad cm 2 (E-F baseline of the SC channel), re-408 spectively, at zero extension for the FDDLs, which corresponds 409 to a contrast loss of less than 1% for the low-spectral-resolution 410 mode of GRAVITY (λ/δλ 22), and negligible contrast losses 411 at medium and high spectral resolutions.This amount of disper-412 sion would be equivalently produced by 14 m of air in the VLTI 413 tunnel using the model for the refractivity of air of Colavita et al. 414 (2004).

Differential dispersion
415 The values for each fiber chain are given in Table 3. Differ-416 ential dispersion varies with the extension of FDDLs as shown in 417 Fig. 6 by approximately 1 µrad cm 2 per millimeter of OPD gen-418 erated by the FDDLs (equivalent to 7 m of air dispersion per 419 millimeter).In the most extreme case, where one FDDL remains 420 at zero while another reaches the maximum stroke of 6 mm, the 421 maximum differential dispersion reaches 8.75 µrad cm 2 (this is 422 the case between beams E and H for example in the Science 423 Channel) leading to an absolute fringe contrast of 94.2%.How-424 ever, this requires a source at the edge of the field at the lowest 425 elevation and in the direction of the longest baseline with the 426 ATs and is therefore very unlikely.In any case, this loss of 5.8% 427 fringe contrast would not prevent fringe tracking compared to 428 a situation with more than 99% fringe contrast, and the loss of 429 contrast in the science channel at such low spectral resolution 430 (no noticeable loss of contrast for spectral resolutions of 500 and 431 4000) would be easily calibrated on an unresolved target.The ef-432 fect of dispersion can therefore be considered close to negligible 433 on fringe contrast.Fiber dispersion needs to be calibrated for 434 astrometry with GRAVITY.This is part of the calibration plan.435 The issue is minimized by using a metrology with a wavelength 436 of 1.908 µm very close to the K band and by measuring the dis-437 persion of the fibers as a function of OPD every month.3. Attenuation of the fiber of the second batch measured by the cut-back method.The peak at 1.9 µm is due to the leakage of high-order modes as the fiber becomes single mode.
Table 3. Differential dispersion and length characteristics of the GRAVITY fiber chains at 240 K and for zero extension of the FDDLs.No error bars are given for the differential dispersion measurements but the minimum value measured with the setup is of the order of 0.1 µrad cm 2 , which gives an idea of the accuracy of the method.The ranges of fringe contrasts in the last column are deduced from these values and obtained by pairing a fiber with the fibers of closest or most different measured differential dispersion in a given channel.The fringe contrast is given for the lowest resolution of GRAVITY (λ/δλ 22).

451
Photons are lost at each of the three connections (Fig. 1) between 452 the Fiber Coupler fiber, the FDDL, the FPR, and the V-groove 453 fiber.Part of the loss is due to Fresnel reflection at the air-glass 454 interface, while another part is due to the misalignment of fiber cores.We used E2000 PC connectors from Diamond to mini-456 mize both effects.The two fiber ends are held in contact with 457 each other in the adapter to minimize Fresnel losses.The fibers are held in a ceramic ferrule encapsulated in a soft metal jacket.

459
The soft metal jacket allows the core-to-core alignment between 460 paired fiber ends to be adjusted very accurately.An accuracy of 461 1 µm is reached with maximum losses of 0.05 to 0.1 dB per con-462 nection or a total additional loss of 3.4% to 6.7% per fiber chain.

463
The third potential main factor is the bending of the fibers in 464 the FDDLs.Losses are all the more important as the wavelength 465 is large as the wave is less confined in the core of the fiber.A 466 first theory was put forward by Marcuse (1976a) where a is the radius of the fiber core and 2α is the bend-loss factor.Marcuse (1976b) computed 2αa as a function of the normalized radius of curvature R/a for a single-mode fiber at the cutoff frequency for values ranging between 300 and 5000.The exponential law can be extrapolated to R/a = 6154 in the case of the GRAVITY fibers yielding a bend loss of 6 × 10 −28 dB for 10 meters of bend length, which is a completely negligible loss.Although the mode is 25% larger at the upper edge of the K band at 2.4 µm, the same conclusion applies at this wavelength.
In conclusion, the losses are in practice dominated by the connections and this is most certainly what the results of Table 4 show with a minimum throughput of 87.5% and an average throughput of 90%, which is excellent given the number of functions performed by the fibers: namely beam transport, polarization control, differential delays, and feeding of the beam combiners.

Raman scattering and fluorescence
GRAVITY uses a 1 W laser at 1908 nm to measure the OPDs between the science and reference beams in each of the four telescopes.The principles of the metrology system are described in Lippa et al. (2016).A small fraction (less than 1%) of the laser light is injected backwards from the two beam combination points and is propagated upward to the telescope entrance pupils to have a full measurement of the OPDs between the science and reference beams in all four telescopes.The high-power fraction (more than 99%) of the laser beam is overlaid with the faint fraction at the entrance of the fiber couplers where the beams coming from the telescopes are fed into the fibers.This threebeam scheme was chosen to minimize the amount of power injected into the single-mode waveguides of GRAVITY.In the initial setup, only the laser beams going through the beam combiners were used, hence with full power going through the fibers.This led to high levels of light back-scattered into the GRAVITY spectrometers because of two distinctive effects that produce additional photon noise in the K band.These effects are described in Lippa et al. (2018).One is Raman scattering, which is the consequence of the interaction between a photon and a phonon.In to be compensated for.We therefore set a goal to reach 6 mm 543 maximum stroke for each FDDL in order to have some margin.

544
The first attempts at fibered delay lines were made by the 545 IRCOM/XLIM group at Université de Limoges in the 1990s 546 with silica fibers.The technique was used to produce a linear 547 optical path modulation, first with a moderate stroke of 20 λ at an accuracy of λ/200 (λ = 633 nm) with a laser control system (Reynaud & Delaire 1993).Zhao et al. (1994) then developed a fibered Mach-Zehnder interferometer at Paris Observatory with standard fluoride-glass fibers, whose optical path was scanned with a fiber wrapped on a piezo cylinder.The stretching of curved fibers introduces birefringence and differential dispersion, both of which have been analyzed (Zhao et al. 1995;Zhao et al. 1995).All these lead to moderate delays of a few thousand λ at most.Simohamed et al. (1996) demonstrated a 318 mm stroke with 20 m of stretched polarization maintaining fibers.Fibers were positioned in 90 • V-grooves with an isotropic distribution of stress in the fiber core in order to minimize induced birefringence.Simohamed & Reynaud (1997) reached a 2 m stroke with 100 m of fibers wrapped on an expanding cylinder.
The relative stretch of a 125 µm fluoride glass was measured to be 0.1% or 2 µm/(mm.N), which therefore allows us to generate several millimeters of delay with sufficient fiber length and sufficient force applied to the fiber.Given the stringent accuracy requirements of GRAVITY, the challenges in building the delay lines were related to the throughput, in particular the potential losses due to the curvature of fibers in the spools, the control accuracy, the birefringence, and the second-order phase dispersion variations as a function of stretch.The characteristics of the FDDLs are listed in Table 1.
The issue of throughput and curvature radius for the spools is addressed in Section 3.4.With a 4 cm spool diameter, no significant loss is to be expected.
The high sensitivity of fibers to mechanical and thermal stresses is a characteristic exploited to build sensors with fiber optics and this sensitivity to physical conditions is an issue when using them for outdoor interferometry, as in aperture synthesis arrays.Fibered delay lines were for example used in 'OHANA primarily as dispersion compensators as the lengths of the fibers were not controlled with a metrology system and suffered from temperature fluctuations along the 300 m of fiber length (Vergnole et al. 2004;Kotani et al. 2005).Following a first experiment by Connes & Reynaud (1988), Reynaud et al. (1992) then showed how to stabilize silica fiber lengths with a laser metrology in order to build a fiber interferometer.A similar technique is used for GRAVITY.To fulfill the specifications on path-length accuracy and time response, the delay lines were designed as two-stage devices: the fibers are wrapped on halfcylinders whose relative distances are actuated by translation stages with servo controlled piezo stacks (modified P-611.1Stranslation stages from Physik Instrument with 100 µm stroke; the control of the piezos is operated with an E-621 controller, the piezo length being measured with a strain gauge sensor) and the overall path-length stability is controlled by the GRAVITY metrology coupled with the GRAVITY piston actuators.This second stage is necessary because of the intrinsic hysteresis of the spools of fibers wrapped around the half cylinders.The motion of the piezo is mechanically amplified by a pantograph whose elasticity is constrained by the loops of fibers.The effect of the expansion of the piezo applied to the pantograph is therefore not linear because of the resistance of the fibers, and the final stroke of the FDDLs is not as large as what is theoretically attainable with the piezo stacks.The control electronics of the FDDLs were designed to ensure that the error of FDDL commands have negligible impact on both fringe contrast and OPD error measurement for typical integration times of several minutes.This requires at least 18-bit precision.For 6 mm of range this translates to a precision of 40 nm per  (FDDL).FDDL E, F, G, H were built with the first generation of GRAVITY fibers while I, J, K, L were built with the second generation of GRAVITY fibers.An optical delay of up to 6 mm is built in both cases.The command is the piezo extension for E, F, G, H while it is a voltage for I, J, K, L to show the amount of hysteresis corrected by the Physik Instrument E-621 servo controller.
FDDL.For long exposure fringes this translates to visibilities 612 of 99% and for OPD measurements to accuracies of 0.6 nm.

613
We designed a real-time controller providing 20-bit precision 614 commands (6 nm precision on a 6 mm range) so that the final 615 performance of our architecture is limited by the performance 6 mm.The E-L delay lines are shown here, while a total of 12 have been produced and labelled A to L. The A-D and E-H were produced first and the I-L were produced later as spares, the difference with the 8 first delay lines being essentially the lowest contamination of the fibers by holmium and thulium.The minimum stroke of 5.7 mm obtained for the F or L delay lines leaves a sufficient margin with which to compensate the maximum bias in OPD of 2.29 mm between beams F and G (Table 3).
Figure 6 shows the amount of second-order dispersion generated by stretching the delay lines of Fig. 5.The second-order dispersion varies for the two batches of FDDLs by approximately 1 µrad cm 2 per mm of generated delay.This additional dispersion reduces the fringe contrast by 5.8% in the most extreme case with a dispersion of R 22 in the K band, as seen in Section 3.3.In practice, this has a negligible effect on the capability of GRAVITY to fringe track and can be easily calibrated.

Fibered polarization rotators
We made the choice to not use polarization-maintaining fibers for GRAVITY, but to use weak birefringence fibers instead, allowing us to avoid systematic splitting of polarizations and there- fore to optimize sensitivity and ensure that GRAVITY is as polarization neutral as possible.The prerequisite is the very low birefringence of the fluoride-glass fibers, which was demonstrated with the very demanding 'OHANA project following the achievements with FLUOR and measured in the case of the GRAVITY fibers (Section 3.2).As the GRAVITY fibers do not maintain polarizations, the axes of linearly polarized radiation can rotate after propagation in the fibers.For weakly birefringent fibers, polarization neutrality can only be achieved if the differential orientation of the axes of polarization can be canceled.
The traditional way to align polarization axes in fibers is to use Lefèvre loops (Lefevre 1980).The basic principle of the device is to induce birefringence in the fibers through stress by bending the fibers.The device is equivalent to wave plates that allow polarizations in the fibers to be controlled.Such devices have been used for FLUOR in a simplified version with two loops only in two perpendicular planes to rotate polarizations without introducing birefringence, as it was found that birefringence was not the primary cause of contrast loss in interferometers with standard fluoride-glass fibers (Perrin et al. 1998).The system was further simplified by Le Verre Fluoré, who later showed that a single loop was sufficient.Such a simplified version was used for FLUOR, VINCI (Kervella et al. 2000), and 'OHANA (Perrin et al. 2006).This is the basis of the system used for GRAVITY, which is described in this section.
The principle of the GRAVITY FPR is quite simple: a torsion is applied to a loop of fiber across a diameter in order to rotate polarizations.The curvature radius is sufficiently large to avoid losses and birefringence induced by stress.An example of FPR is presented in Fig. 7.A shaft is actuated by a stepper motor to twist the fiber along the vertical axis.The fiber is loose in the hole at the top of the shaft and at the input and output feedthroughs.End switches allow the twist range to be controlled in order to avoid damaging the fiber.

769
After the polarizer, the wave is projected on the axis of the po-770 larizer − → e P and is written as: The wave is then injected into the fiber and propagates down to the analyzer (we assume the coupling to the fiber is lossless).

773
The wave accumulates birefringent phases on the x and y axes.

774
At the fiber output, this can be written as: The average intensity detected at the output of the analyzer is 780 independent of z and t and is written as: The first term takes different values depending on the nature of 782 the input wave; for an unpolarized wave, it is equal to 1 2 I np , 783 while for a polarized wave it is equal to I p cos 2 (ψ − θ P ).This where the first line is the unpolarized case and the second one is the polarized case.The cosine term in the square parentheses is the classical Malus law for polarized light and the second term is the effect of birefringence.We note that in the case of polarized light, there is a cascade of two Malus laws as the light filtered by the two polarizers is already polarized in the input.

Appendix B: Measurement of birefringence
We show in Appendix A that the classical expression of the Malus law needs to be modified to account for birefringence.
Conversely, it can be used to measure the birefringence properties of the fiber.Natural light is used to measure the properties of the GRAVITY fibers.In the following, we therefore consider the first line of Eq.A.7 for unpolarized light.It is a family of curves with parameters either θ A or θ P with θ P or θ A being the respective variable.We show here that the characteristics of the envelope of this family depend on the birefringence phase δϕ.
In the following, we consider θ P as the parameter and θ A as the variable.We highlight the fact that Eq.A.7 is symmetric and therefore either case can be chosen.The expression of the envelope of the curves is classically deduced by expressing that it both takes generalized Malus law values and is tangent to the curves.It amounts to solving the system of equations: + sin 2 δϕ 2 1 + e 2i(θ A +θ P ) .
Defining c A = cos(2θ A ), s A = sin(2θ A ), c P = cos(2θ P ), s P = sin(2θ P ), γ 2 = cos 2 δϕ 2 , and σ 2 = sin 2 δϕ 2 , the above system can be written in matricial form as: and can be solved for c P and s P .The matrix cannot be inverted if θ A = π 4 ( π 2 ) and ϕ = π 2 (π), in which case I env = 1 4 I np .Otherwise the system can be inverted, yielding values for c P and s P leading to a quadratic equation by expressing: c 2 P + s 2 P = 1.The quadratic equation has two solutions in the general case leading to upper and lower envelopes: These expressions depend on the birefringence phase δϕ and provide a method to measure it.Some examples for various birefringence differential phases are plotted in Fig. B1.In the particular case of δϕ = 0(π), I low and I up are constant, and are equal to 0 and 1, respectively.In the absence of birefringence, the generalized Malus laws systematically reach the maximum amplitude and in the case where the fiber length is a multiple of half of the beat length of birefringence, birefringence has no effect on the generalized Malus law.When birefringence is small, as in the   and can be used to measure it and predict its effects on the mea-833 surements made with GRAVITY.

1
GRAVITY is a second-generation instrument of the Very Large 2 Telescope Interferometer (VLTI) of the European Southern Ob-3 servatory (ESO).It is operated in the K band between 2 and 4 2.4 µm and allows long-baseline interferometry with either the 5 four 8 m Unit Telescopes (UTs) or the four 1.8 m Auxiliary 6

Fig. 2 .
Fig.2.Example of generalized Malus laws for 18 orientations of the polarizer upstream from the injection in a chain of fibers made of FDDL A, FPR A, and the A fiber coupler (total length of 18.88 m), and 18 orientations of the analyzer downstream with the setup of Fig.A1.The dashed line is the envelope of Eq.B1 with a measured birefringence phase difference between the two polarization axes of 20 • , yielding a birefringence beating length of 339 m according to the method presented in Appendix B.
Differential dispersion is taken here to mean the accumulated difference of the second-order variation of the phase of the wave as a function of wavenumber between two fibers caused by the propagation in the fibers.Namely, φ (σ) if the interferogram measured by GRAVITY writes A+B cos (2πσx + φ(σ)), where x is the OPD.The differential dispersion term is measured in units of µrad cm 2 .The effect of fiber dispersion is to spread the interferogram over the OPD.More specifically, the effect of the first derivative of the phase is to shift the interferogram by − 1 2π φ (σ) and the secondorder derivative yields an OPD shift as a function of wavenumber.As a consequence, interferograms at different wavelengths will not contribute to the maximum-contrast white-light fringe at the same OPD, therefore reducing contrast in wide band.The other consequence is that measuring the fringe phase, which contains the intrinsic visibility phase information, requires a calibration.Intrinsic fiber dispersion cannot be canceled but differential dispersion can be minimized.Differential dispersion has several components: material dispersion due to the dependance of refractive index on wavelength, waveguide dispersion due to the dependance of the mode on wavelength, and profile dispersion due to the variation of refractive index along the fiber (Coudé du Foresto et al. 1995).In addition, geometric irregularities along the fiber also add a dispersion term.The first two terms dominate and have opposite signs.Differential dispersion is classically compensated by increasing the fiber length of one arm relative to the other in the two-arm interferometer at the expense of optical-path equalization, which is achieved with the FDDLs in the case of GRAVITY.The differential dispersion of the 'OHANA fluoride glass-fiber cables was minimized by combining six sections of 50 m fibers with compensating effects (Kotani et al. 2005; Perrin et al. 2006).The same technique has Fig.3.Attenuation of the fiber of the second batch measured by the cut-back method.The peak at 1.9 µm is due to the leakage of high-order modes as the fiber becomes single mode.
440 tors; first of all, by the intrinsic transparency of fluoride glass 441 fibers.The attenuation by the fibers used for GRAVITY is shown 442 in Fig. 3 as measured with the cut-back method on the last batch 443 of fibers produced for GRAVITY.The attenuation increases to-444 wards the blue because of Rayleigh scattering and increases af-445 ter 2.3 µm because of absorption by water.The attenuation has a 446 local peak at 1.9 µm because of the suppression of high-order 447 modes as the fiber becomes single mode.The attenuation is 448 6.63 dB/km at 2 µm and 3.65 dB/km at 2.4 µm, yielding trans-449 missions of respectively 97% and 98% for 20 meters of fiber.450 Connections are another possible source of transmission loss.
assuming infi-467 nite radius for the cladding.A more realistic approach has been 468 proposed to account for the limited radius of the cladding (Faus-469 tini & Martini 1997), even with metallic claddings (Peng et al. 470 2017).The bend loss in dBs for a given curvature radius R and 471 length of bend L can be written (adapted from e.g., Peng et al. 472 (2017)): 473 L s = 10 log 10 exp 2αa L a ,

Fig. 4 .
Fig. 4. Pair of GRAVITY FDDLs.Four blue fiber jackets are visible at the top and bottom of the box.The fibers are wrapped on two half cylinders, one fixed and one mounted on a piezo actuator.Delay is produced by stretching the fibers with the piezos.The gray and white cables on the left and right are the power and gauge sensor cables of the piezos.

Fig. 5 .
Fig.5.Examples of optical path-length laws for two batches of Fibered Differential Delay Lines (FDDL).FDDL E, F, G, H were built with the first generation of GRAVITY fibers while I, J, K, L were built with the second generation of GRAVITY fibers.An optical delay of up to 6 mm is built in both cases.The command is the piezo extension for E, F, G, H while it is a voltage for I, J, K, L to show the amount of hysteresis corrected by the Physik Instrument E-621 servo controller.

Fig. 6 .
Fig.6.Examples of second-order differential dispersion laws for the two sets EFGH and IJKL of FDDLs.The average dispersion was set to zero at minimum FDDL extension.Both sets of delay lines generate a maximum of 5 µrad.cm 2 at a maximum extension of 6 mm.Abscissas are as in Fig.5.

Fig. 7 .
Fig. 7. GRAVITY fibered polarization rotator.The blue jackets are the fiber output and input.The loop of fiber goes through a hole at the top of the shaft, which defines the rotation axis.The rotation of the shaft is controlled by a stepper motor.The two mechanical end switches are visible at the bottom of the box.The torsion applied to the fiber by the rotation of the shaft generates a rotation of the polarization of the beam going through the fiber.

Fig. 8 .
Fig. 8. Examples of Malus laws (top) and angle laws (bottom) for the fibered polarization rotators IJKL.The angle has been arbitrarily set to 0 o for 0 motor steps.
θ P , z, δϕ, t) = e −i(kz−ωt) cos θ P E x + sin θ P E y × (A.4)e −i δϕ 2 cos θ P − → e x + e i δϕ 2 sin θ P − → e y .The wave is filtered at the output of the fiber with an analyzer 776 rotated by an angle θ A with respect to the − → e x axis.The effect 777 of the analyzer is to project the incoming wave in the direction 778 − → e A = cos θ A − → e x + sin θ A − → e y , giving the analyzed wave: 779

Fig. A1 .
Fig. A1.Setup used to measure the generalized Malus laws of a fibered device as a function of input linear polarizer angle θ P and output analyzer angle θ A . 832

Table 1 .
Main characteristics of the fibers and Fiber Control Unit elements of GRAVITY.More details can be found in the dedicated sections of this paper.

Table 4 .
Differential length (m) Differential dispersion (µrad cm 2 ) Min-max contrast (%) Throughput of the GRAVITY fiber chains.Each chain comprises the Fiber Coupler fiber, an FDDL, an FPR, and a V-groove fiber.The accuracy on the throughput measurements is of the order of 0.1%

a
Programmable Logic Controller from Beckhoff.Motion com-696 mands are sent in units of micro-steps: eight micro-steps per real 697 motor step.One full motor revolution is accomplished in 200 698 steps so that a full revolution requires 1600 micro-steps.This 699 provides ample resolution to accurately rotate the polarization 700 by 0.25 • per micro-step.The motor is turned off when not in use 701 by setting the hold current to zero to avoid heating in vacuum 702 and errors due to fluctuating power.The FPRs lie in a horizontal 703 plane so that no mechanical motion can be set by gravitation.The performance of the FPRs was measured at 240 K. Malus 705 laws were measured by injecting linearly polarized light into the 706 FPRs and detecting the rotated polarization with an analyzer at 707 the output.Examples of resulting Malus laws are presented in 708 Fig.8.The rotation angle versus motor step command laws can 709 be easily deduced from the Malus laws and are presented in the 710 same figure.From these, the goal is achieved with an amplitude 711 of rotation of larger than 180 • .The amplitude could still be in-712 creased by moving the end switches slightly.The rotations are 713 reproducible with an accuracy of the order of 1 • or better (8 • is 714 necessary to reach 99% fringe contrast).In practice, the FPRs are 715 used to align the linearly polarized light of an internal calibration 716 source(GRAVITY Collaboration et al. 2023) which ensures a 717 high level of visibility contrast on sky (GRAVITY Collaboration 718 et al. 2017).The FPRs are not operated during the observation 719 nights and remain at fixed positions.usedto analyze the properties of the fibers is described by a P is the orientation of the polarizer axis with respect 704 761Here, θ 767 shifted to have opposite amounts of birefringent phase on the x 768 and y axes).