Accurate interferometry underpins many high-precision measurements ranging from, e.g., gravitational wave detection [1], over Earth rotation sensing [2] to volume determination of silicon spheres [3,4] for the revised SI kilogram [5]. In optical frequency transfer via interferometric fiber links (IFLs), accurate interferometry enables path length stabilization [6–9] achieving frequency transfer uncertainties at or below the 10−19 level over IFLs having a length >500km [10–13]. Hence, IFLs are currently the only available means to remotely compare [14] the best optical clocks. These now reach systematic uncertainties below 3×10−18 [15–21]. Remote optical clock comparisons are not only considered as a prerequisite for a potential redefinition of the SI second [22,23] but also enable applications in chronometric leveling [15,20,24–26] or fundamental physics [27,28]. Despite the proven frequency transfer performance, the interferometric noise floor [8,9,29] was identified as limiting uncertainty contribution at averaging times ≥1000s in recent assessments [12,13,30]. Further progress of IFL performance and study of remaining non-reciprocal effects [12,31] require an improved interferometer.

Uncompensated fiber paths, which are the source of interferometric noise, can be located either at the sender site or in the receiver at the remote end. A typical example at the sender end would be the (short) reference arm of the Michelson interferometer of which the IFL forms the long arm. At both the sender and the receiver ends, input and output fibers are often uncompensated. In remote clock comparisons [14,15,25,32], a single IFL is typically only one constituent in a longer frequency transfer chain [33]. In this case, it is sometimes possible to avoid uncompensated paths altogether, by ensuring that the input and output reference planes of successive IFLs coincide [34,35]. In practice, however, many setups, including commercially available equipment, still contain uncompensated paths. In some topologies, common-mode suppression between multiple uncompensated paths exposed to the same environment can be utilized to minimize the net interferometric noise contribution [36,37].

For assessing the frequency transfer performance—online in an optical clock comparison as well as for validating IFLs and their instrumentation, one typically implements an out-of-loop (OOL) characterization by looping back the light from the remote end to the sender and comparing the loop output signal to the input of the IFL. The OOL setup typically introduces additional interferometric noise, impacting the assessed frequency transfer uncertainty. Ideally, this OOL signal would characterize the complete frequency transfer error between the IFL sender input and receiver output. However, the loop output signal typically bypasses part of the receiver setup and, therefore, this part of the frequency transfer error cannot be assessed. Worse, because the sender and the comparison setup reside at the same location, uncompensated paths are now exposed to the same environment and noise contributions may cancel, falsely suggesting a lower uncertainty than will be achieved for separated sender and receiver units. Nevertheless, the best one can do for a meaningful OOL characterization is to minimize uncompensated paths in the sender and comparison setup and avoid their mutual cancellation. Furthermore, it is important to ensure that different contributions do not mask each other, i.e., that the frequency transfer uncertainty assessed via the OOL signal is an upper limit to the frequency transfer error of the signal received at the remote end.

Here, we demonstrate a compact all-dielectric free-space interferometer, as an accurate yet easily implementable OOL characterization solution. Different from a recent implementation of a compact free-space interferometer [37], the central element is a monolithic assembly formed by optically contacting a polarizing beam splitter (PBS) with a quarter-wave plate (QWP) and a high-reflection (HR) mirror. We integrate this assembly into a compact OOL characterization setup and we report OOL fractional frequency uncertainties of 10−21 and below.

In the following section, we will elucidate the relevant principles of IFLs and introduce the terminology used throughout the paper. After that, we will continue with introducing and discussing our interferometer design in Section 3 and presenting the experimental results in Section 4. In Appendix A, we address strategies to further enhance the achieved performance in the future.

Figure 1(a) shows the generic setup used for the fiber-based, stabilized frequency dissemination to a remote location depicted here as being implemented using fiber optics.

The path length stabilization relies on the retro-reflecting part of the light reaching the remote end back to the sender. In this way, the IFL forms one arm of an unbalanced Michelson interferometer. The round-trip phase evolution of the long arm to the remote end is stabilized with respect to a short reference arm using a phase-locked loop (PLL) actuating an inline acousto-optic modulator (AOM). A second AOM near the remote end serves to discriminate light reflected at the remote end from spurious reflections and backscattering along the IFL. Within the locking bandwidth, the PLL establishes phase coherence between the reference arm retroreflector (place A) and the remote end B. If propagation is reciprocal, i.e., the phase evolution is equal in forward and backward propagation directions, stabilization of the round-trip phase offset implies phase stabilization of the one-way signal [8]. Suppression of the fiber connection phase noise is constrained by the so-called delay limit [8], leading to a dynamic residual frequency transfer error Δfctrl.

In the following derivation of a mathematical treatment of the OOL signal, we assume that the purpose of the IFL is to transfer the optical frequency νC1 of the light wave incident from an ultrastable laser to the coupler C1 to the remote place B′. This choice of the coupler C1 as starting point simplifies the following argumentation since it is the splitting point between the long-haul fiber and reference arm. The treatment can easily be adapted to different choices of the starting point. Note that in some applications it is only required to achieve phase coherence between two locations physically different from the IFL input.

As elucidated above, the phase stabilization acts with respect to the short reference arm of the Michelson interferometer. Since this reference arm is not free of variations, the IFL essentially does not transfer the frequency νC1 but the frequency νA=νC1+ΔfC1−R−C1/2,(1)with ΔfC1−R−C1 being the frequency error introduced by the reference arm. The factor 1/2 stems from the action of an idealized control loop within the stabilization bandwidth. For short, we will call this frequency error ΔfC1−R−C1/2 the in-loop (IL) contribution to the interferometric noise. Here and below, we make use of the definition of the frequency shift introduced by propagating from X to Y with ΔfX−Y(t)=ddt∫0LXYδϕ(z,t−(τXY−z/cn))dz,(2)where cn is the speed of light in the fiber, LXY is the geometric path length between the points X and Y, τXY=(zY−zX)/cn is the corresponding propagation delay, and δϕ is the phase noise along the considered fiber path. We omit temporal variations of LXY and τXY for the sake of simplicity. Furthermore, we introduce a short-cut notation for a concatenated path: ΔfX−Y−Z(t)=ΔfX−Y(t−τYZ)+ΔfY−Z(t).(3)

Similarly, there is a frequency error introduced at the remote end which results from the differential phase variation between the point B, which is the mid-way point actually stabilized by the IFL, and the location B′, where the transferred frequency signal is effectively used: ΔfB′,B=ΔfC2−B′−ΔfC2−B.(4)

Hence, assuming reciprocal propagation, the frequency transfer error Δf=νB′−νC1 in stabilized operation can be written as Δf=Δfctrl+[ΔfC1−R−C1/2]Es+[ΔfB′,B]Er,(5)where [·]Ex specifies the environment driving the interferometric noise terms (Es is the sender site and Er is the remote site). Relevant environment variables are those that change the specific optical path length, for example, temperature, (air) pressure, and humidity; see, e.g., Ref. [34] for a detailed discussion.

As Eq. (5) shows, an accurate one-way frequency transfer requires minimizing ΔfC1−R−C1 and matching the frequency disturbances on the paths C2−B and C2−B′, i.e., to minimize ΔfB′,B. The mutual cancellation of disturbances has been discussed in the context of improving many point frequency extractions [38] and repeater laser stations [29]. In fiber optics this can be achieved by length matching the fiber leads after the fiber coupler and routing them close to each other in effectively the same environment [9,29,30,38]. Alternatively, the contribution C2−B can be completely avoided by using a partial retro-reflector [7]. This is optimally placed at the user output B′ to remove the C2−B′ contribution. A partial reflector can also be used at the sender instead of C1 and R; however, this leads to unwanted multiple reflections into the link which can cause trouble, especially with amplifiers that amplify true and spurious signals equally.

The loop-back configuration used for the OOL characterization of the IFL performance is shown in Fig. 1(b). While for the pure assessment of the IFL performance or its instrumentation a direct loop-back is often used, applications require to tap off the signal at the remote end, e.g., via a repeater laser station [11,29] [Fig. 1(c)] or a manypoint extraction [38] [Fig. 1(d)]. Uncompensated paths between the tapping point on the IFL and the effective user output are not monitorable by looping back and need to be taken into account in the construction and design phase of the extraction setups [11,29,38]. Here, we focus on the OOL signal and characterization of the contributions of sender-side imperfections on the basis of the setup shown in Fig. 1(b).

The observed frequency error in this OOL characterization in stabilized operation can be written as $$\begin{gathered} ΔfOOL \\ =Δfctrl+[ΔfC1−R−C1/2+ΔfB′,B+ΔfB′−C1]Es \\ =Δfctrl+[ΔfC1−R−C1/2−ΔfC2−B+Δfconn]Es, \end{gathered}$$(6)with the optical path length variations of the connection used to generate the OOL beat being Δfconn=ΔfC2−C1.

Equation (6) shows that an additional contribution ΔfB′−C1 limits the resolution of the OOL characterization. Furthermore, the rewrite in the last line of Eq. (6) points to an unwanted artifact in IFL OOL characterizations: common mode cancellations of the three terms ΔfC1−R−C1/2, ΔfC2−B, and Δfconn can arise from being immersed in effectively the same environment Es. While such a common mode rejection is beneficial for a more stringent assessment of Δfctrl [9], it limits the significance of the OOL characterization as a measure of the frequency transfer uncertainty to remote locations since the fluctuations ΔfC1−R−C1/2 impacting the one-way frequency transfer are not completely visible in the OOL signal.

The impact of the Δfconn contribution to the OOL signal can be avoided by implementing a two-way comparison between the loop output and the input [39,40], which typically requires an additional AOM in the setup shown in Fig. 1(b). However, even for a two-way comparison, the two uncompensated paths between the coupler and retro-reflector at both sides remain as interferometric uncertainty contributions. Here, we discuss a different strategy for enhancing the OOL resolution limit based on minimizing the environmental sensitivity of ΔfC1−R−C1/2 and Δfconn together with rendering ΔfC2−B=0 by inline retro-reflection. This setup makes use of passive optical components only and does not require any two-way post-processing.

The interferometric noise floor is assessed by short-cutting the IFL rendering Δfctrl negligible: ΔfOOL,sc≃[ΔfC1−R−C1/2−ΔfC2−B+Δfconn]Es,(7)which defines the baseline for the OOL characterization signal from Eq. (6). The interferometric noise floor depends, as discussed above, on implementation details. These are not always described in publications or even known to the full extent rendering the comparison of results from different groups intricate. Table 1 summarizes published interferometric noise floor results together with the details mentioned. Reported fractional frequency instabilities of the interferometric noise floor are compared in Fig. 2.Table 1.

Summary of Literature Interferometric Noise Floor Results Together with Disclosed Configuration Details and a Qualitative Classification^a

Short-cutting the IFL not only removes the delay-limited phase noise contribution of the long-haul IFL but also changes the transfer function of the control path [8,42]. Hence, phase noise suppression features are different between short-cut configuration and long-haul IFL. Furthermore, other quantities such as the polarization variation dynamics may vary resulting in different frequency error contributions [8,43]. These effects need to be considered separately.

Our approach to simultaneously minimize the interferometer-induced contributions to frequency transfer uncertainty and OOL characterization uncertainty is illustrated in Fig. 3. We use a compact, free-space, all-dielectric interferometer layout. The central unit for path length stabilization is a monolithic assembly consisting of three optically contacted elements (manufactured by Layertec GmbH): a PBS made from fused silica and having an edge length of 10 mm, a low-order QWP made from a single quartz substrate (thickness 227 μm), and an HR mirror.

The rationale behind choosing optical contacting as the joining method instead of gluing is to avoid long-term creep of adhesives and their greater sensitivity toward changes in the environment.

The source light incident on the PBS is split up in its polarization components. The s-polarized portion is directed to the HR mirror and returns back to the PBS reflection plane p-polarized due to the action of the QWP. This path forms the reference arm of the Michelson interferometer. The p-polarized part of the incident light is transmitted through the PBS and focused into a single-mode IFL fiber, where it passes through a Faraday rotator (FR), two AOMs, and a polarization controller. At the far end of the IFL, the light beam hits a partially reflecting (PR) mirror terminating the IFL [7,30]. The polarization of the wave transmitted through the PR mirror is set to be linear. By doing so, the retro-reflected light propagates the fiber section backwards and due to 90° net polarization rotation by the FR double passage it is reflected by the PBS. The 90° net polarization rotation could also have been implemented using a QWP. After polarization projection, the beating with the reference arm light is recorded using a free-space photodiode. For the OOL characterization, the IFL output light is transmitted through the PR mirror and is superposed with a fraction of the input light beam tapped off by a non-polarizing beam splitter (NBS). Initially, we tried to optically contact the externally sourced monolithic IL assembly, the NBS, and the PR mirror to each other. Unfortunately, the achieved optical alignment was not sufficient so that in the following we kept the NBS and the retro-reflector separate and aligned them individually. The NBS is made from N-BK7, has an edge length of 10 mm, and is placed at a distance of approximately 3 mm to the monolithic assembly. Again, the beating is recorded after polarization projection with a free-space photodiode.

In order to avoid metallic optics mounts [8] with their larger thermal expansion, the monolithic assembly and the NBS are glued onto a fused silica substrate (diameter 50 mm, thickness 10 mm) by placing a fillet made from room temperature vulcanizing silicone at the free edges of the contact area between the beam splitter and substrate. This gluing procedure is inspired by other applications requiring a stress-less fixation of optical elements [44,45]. In the same manner, the substrate is glued onto an optical bench (Thorlabs, FT-100X100) used for fiber coupling.

We decided to base our setup upon beam splitter cubes for their standard availability and their ease in assembling the setup on commercially available fiber benches. The surfaces of these beam splitter cubes are anti-reflection coated in order to minimize potential etalon effects.

Two different positions of the PR retro-reflector have been realized. In a first version abbreviated as S1, we used an inline retro-reflector coated onto the fiber end resulting in a spatial separation of ≈37mm between the PR mirror and NBS edge, i.e., a large OOL path length. Later we used an uncoated fiber and glued a PR mirror at a distance of ≈1.5mm to the NBS edge using a UV curable epoxy (Optocast 3410 Gen2). This configuration has been characterized by an IFL consisting of single-mode fibers—in the following named S2, as well as using polarization-maintaining fibers for the IFL—identified as S2PM in the remainder.

The setup was placed on an optical table in a climatized laboratory. The servo and marker AOM drive frequencies have been −55MHz and 40 MHz, respectively, for the S1 and S2 configurations resulting in IL and OLL beat frequencies of ≈30MHz and ≈15MHz. In the S2PM the IL and OOL beat frequencies have been lowered to ≈12MHz and 6 MHz, respectively, using polarization-maintaining AOMs with drive frequencies of −74MHz and 69 MHz. The fiber length of the characterization IFL in both setups has been <5m, giving a negligible self-heterodyne phase noise contribution [8] induced by the phase noise of the narrow-linewidth fiber laser source with an operating wavelength of ≈1542nm. The photodiode signals have been recorded with dead-time free counters (K+K FXE), which have been set to report 1 s phase averages of the frequency readings collected at the internal gate time of 1 ms. This type of phase averaging is also called Λ-averaging [46,47].

In the configurations S1 and S2, the polarization was manually optimized before the measurements. The following characterization concentrates on the configuration S2PM, as this avoids polarization effects to enter the assessment of the interferometric path length noise characterization. We assume that a performance similar to the characterization below can be achieved for non-polarization-maintaining fiber connections with stronger birefringence changes when using existing polarization stabilization techniques [48,49].

In this interferometer layout, the noise term ΔfC1−R−C1 in Eqs. (5) and (6) is given by the beam path between the PBS reflection plane and the HR mirror resulting in a geometric reference arm length of ≈2×5.2mm for centric transversal. Furthermore, ΔfC2−B=ΔfB′,B=0 due to inline retro-reflection. Accordingly, in our setup Δfconn is induced by the optical path between the inline retro-reflector and the PBS reflection plane: Δfconn=ΔfB′−C1. In the following paragraph, we will quantify the temperature and air pressure sensitivities based on the properties of the involved materials.

Variations in environmental variables χ induce a fractional frequency error given by [29,34] $$\begin{gathered} y(t)=1c0∑i(∂ni∂χiLi+ni∂Li∂χi)dχi(t)dt \\ =∑iγχidχi(t)dt, \end{gathered}$$(8)where the summation takes into account different sections i of the considered optical paths with the refractive index ni and geometrical length Li, where c0 is the speed of light in vacuum. As quantities χi, we will analyze and measure the impact of temperature (T) and air pressure (p) variations in the following. We will assume equilibrated environmental variations in the interferometric setup χi(t)=χ(t) which we consider justified due to the compactness and the shielding of the setup.

Using Eq. (8) and the material properties listed in Table 2, we calculated the environmental sensitivities γχ=∑iγχi of the different paths listed in Table 3. In calculating these sensitivities, we assumed centric transversal of the optics and modeled variations of the geometric lengths to act on the center-to-center distances. Furthermore, for describing the action of the QWP, we used the mean of ordinary and extraordinary refractive indices, the thermo-optic coefficient, and the pressure dependence of the refractive index. The impacts of air paths are described using an updated version [55] of the Edlén formula [57,58] and their partial derivatives with respect to T and p. We evaluate these at the working point derived from the barometric formula for a height of h=77m, a gravity constant g=9.81ms−2, in standard atmosphere defined by the international civil aviation organization [air density ρ0=1.225kg/m3, air pressure p0=1013.25hPa, relative humidity 0%, CO2 concentration 450 ppm (parts per million)], giving p=1004.04hPa, and at the laboratory temperature of T=20°C. For the configurations S2 and S2PM, we calculate [55,56] a sensitivity of −0.13as/% relative humidity variation and we will see below that the resulting effects are negligible compared to the impact of T and p fluctuations.Table 2.

Materials Properties Used for the Calculation of the Environmental Sensitivities (Operating Conditions: λ=1542nm, T=20°C, p=1004.04 hPa, 0% Relative Humidity, 450 ppm CO₂ Concentration)^a

The T and p sensitivities effective in OOL characterizations are obtained by adding the ΔfC1−R−C1 and Δfconn sensitivities in the respective configurations yielding γT,OOLS1=1.86fs/K,γp,OOLS1≈31.86as/hPa,(9)γT,OOLS2=0.51fs/K,γp,OOLS2≈4.00as/hPa.(10)

As the sensitivities of the individual paths given in Table 3 show, the impact of T and p variations, considered separately, will only add up for the different paths. Hence, the OOL characterization signal overestimates the frequency error of the signal extracted at the remote end for pure T or pure p variations. A spurious mutual cancellation of noise terms in the OOL characterization is only possible by the combined impact of T and p variations. We will see below, however, that for the improved configuration S2 and for the interesting averaging times where the OOL signal starts to flicker, the typical estimated impact of T dominates over p up to averaging times of ≈3×104s. Table 3 also shows that the p sensitivity mainly results from air paths of the free-space setup. By comparison, the single-mode fiber Corning SMF28 exhibits a temperature sensitivity of 37 fs/(K m) [29]. This shows that the IL and OOL T sensitivities are equivalent to uncompensated fiber paths of only 0.5 cm and 1.4 cm in length, respectively.

Figure 4 compares the instabilities in the different configurations to the impact of T and p variations estimated on the basis of Eqs. (9) and (10) and simultaneous T and p measurements. Figure 4(a) for the S1 configuration shows that for averaging times >1000s the observed OOL instabilities (red) match the estimated impact of T (green) and p (blue) variations well. In this configuration, we observe a crossover between being T variations limited, for averaging times between 103s and 104s, and being p variations limited for averaging times ≳104s that is reflected in the OOL instability. From this finding and the close match between the measured and estimated temporal OOL phase evolution shown in Fig. 5, we deduce that the modeling of the environmental sensitivities approximates the reality to a high degree. The residual impact of T and p variations leads to a stagnation of the observed OOL fractional frequency instability at a level of ≈4×10−21 in the S1 configuration.

For the configurations S2 and S2PM, however, the impact of p variations is greatly reduced due to the shortening of the air path between the retro-reflecting mirror and the NBS. For these configurations we observe OOL fractional frequency instabilities below 10−21 for averaging times above 5×104s. For the S2PM configuration, we have achieved a significantly reduced short-term instability at averaging times <100s. While switching from the S2 to the S2PM configuration, we observed that this reduction mainly resulted from using a PM fiber connection between the source laser and the entrance of the interferometer setup. An additional improvement of the short-term instability below averaging times of 4 s has been achieved by enhancing the signal-to-noise ratio of the beats. The S2 and S2PM measurements also show a larger discrepancy from the estimates of the impact of T and p variations than the S1 measurement. This is observable in the instabilities in Figs. 4(b) and 4(c) and also in the time traces (not shown), which hints to uncharacterized OOL processes not described by the material response (see discussion below).

In the S2PM configuration we have performed several measurement runs to analyze the statistics of its performance. Figure 6(a) shows the observed instabilities and compares these to the results of Akatsuka et al. [36], Cantin et al. [37], and to the instability contribution of the stabilization electronics. The instability of the OOL signal displays a reproducible behavior for averaging times <100s, with a spread in the results below a factor of 2. Measurements with shorter gate time (not displayed) show a modADEV progression proportional to τ−3/2 for averaging times between 10−3s and 10−1s and a shoulder-like contribution setting in around 1 s in configuration S2PM (≈0.5s for configuration S2). The OOL phase noise density in Fig. 6(b) (Symmetricom 5125A) shows that the latter contribution results from a steep phase noise increase below 1 Hz. We checked that neither the tracking filters [orange trace in Fig. 6(b)] nor the combination of analog phase-frequency comparator and proportional-integral controller [gray trace in Fig. 6(b)] is responsible for this phase noise contribution.

For longer averaging times, the OOL instabilities in Fig. 6(a) again exhibit a stagnation at a level understandable by the T variations (green curves). In the future, an improved passive and active T stabilization may enhance the results in this range. For averaging times between 103s and 5×104s, the estimated p-induced instabilities (blue curves) are approximately a factor of 10 below. This shows that a mutual complete cancellation of T and p is not present in these measurements.

The OOL instabilities we obtain are up to averaging times of 200 s below the best reported interferometric OOL characterization noise floors so far. For longer averaging times, only the interferometric noise floor of the multi-branch frequency distribution systems [36,37] shows comparable low instabilities: while the result of Akatsuka et al. [36] shows an instability at the lower edge of the range we observe, the result of Cantin et al. [37] levels off due to T variations a factor of ≈5 below our average performance. Both results, however, are essentially the inter-branch comparison of compact interferometric ensembles designed to maximize the inter-branch common-mode noise rejection [36,37] and, hence, quantify the uncertainty of the multi-branch frequency distributions system. When a pair of those branches are used for dissemination and OOL characterization, this inter-branch comparison signal does not necessarily quantify the frequency transfer error to a remote place at the same uncertainty level due to common-mode rejection. Interestingly, the instability Cantin et al. [37] report after compensating for the measured T variations falls significantly below the estimated impact of p variation on our 4.5 mm long OOL air path. Assuming a similarly long air path of the interferometric ensemble and expecting long-term p variations to be comparable in Paris and Braunschweig, this also indicates a high degree of common-mode rejection.

Figure 7 shows the OOL fractional frequency transfer uncertainties in several measurement runs for Π and Λ averaging. For 25 out of the 28 measurement runs shown in Fig. 7(a), the observed fractional frequency uncertainties are below 2.7×10−21 with the offsets exceeding 1.0×10−21 in only three (Π-averaging) and two (Λ-averaging) cases (see dotted line). The observed OOL frequency offsets are also compatible with zero within 1σ (σ being the statistical uncertainty contribution) for 27 (Π-averaging) and 25 (Λ-averaging) out of the 28 measurement runs. The observed offsets are therefore roughly a factor of 4 better than previously published inaccuracies (see Table 1).

Figure 7(b) shows the residual uncertainties after compensating for the estimated impact of T and p variations. These residual uncertainties scatter on a lower level with 22 of the 28 runs being below 1.0×10−21 (Λ-averaging). The zero-compatibility is as distinct as for the analysis without T and p compensation. The 1/σ2-weighted average of all measurement runs in S2PM configuration is (0.23±1.07)×10−22 and (2.18±8.53)×10−23 for Π and Λ averaging, respectively. After compensating for the estimated impact of T and p variations, these weighted averages are (1.37±8.60)×10−23 and (1.08±6.99)×10−23, respectively. This shows, as the scattering of the individual runs, that the overall uncertainty is reduced by accounting for the T and p variations.

However, this reduction is not pronounced, which indicates that there are uncharacterized OOL processes with a similar magnitude present in our system. Free-space interferometers are susceptible to uncertainty contributions resulting from optics movements and related beam pointing variations. A full description of this effect requires integrating the electric field interference of the two heterodyned beams over the photodiode surface [60] including their spatial wavefronts. In our setup, the focal length of the fiber collimator used is 7.5 mm (Thorlabs, PAF-X-7-C) resulting in a collimated beam diameter of ≈1.4mm for the 10.4 μm mode field diameter of the prototypical Corning SMF28 fiber. For such a collimation, the Rayleigh range is zR≈1.0m. Hence, we expect the wavefront curvature and the Gouy phase shift to play a lesser role and expect the geometrical contributions to be dominated by mutual longitudinal shifts of the beams and angle effects. In the case where longitudinal shifts are assumed to be the only source of phase disturbances, the offsets we observe would correspond to ≤150nm beam displacements, which does not seem to be out of range given the mounting of optics in our setup. Such residual OOL errors may also result from movements of the retro-reflecting mirror, e.g., by changes of the UV curable adhesive underneath. Indeed, a comparison of the observed temporal phase variations with the ones estimated by T and p variations (not shown) indicates that often the estimated T variations do not completely account for the observed OOL phase error and that on average a factor of ≈2.2 is required to minimize the discrepancy. Further studies have to show whether different adhesives or making the setup more compact by fixing the NBS, PBS, and miniaturized fiber couplers to the same baseplate leads to a reduction of such OOL processes.

A different OOL process may result from air humidity h variations in the air paths, which we have not addressed so far. By combining the above-mentioned relative humidity sensitivity of −0.13as/% with the typical relative air humidity variation measurements in the same laboratory [34], we estimate the uncertainty contribution of these effects to be ≤5×10−23 for averaging times between 103s and 105s. Hence, air humidity variations can be ruled out as the source of the observed discrepancy.

Furthermore, ghost beams as, e.g., discussed in the gravitational wave detection community [61–63] may contribute to the OOL signal. In our setup relevant ghost beams have to be polarized orthogonal to the outgoing signal. Hence, residual reflections of AR coatings in the linearly-polarized beam path are of lesser importance. However, there are two relevant ghost beam sources: first, backreflections or Rayleigh backscattering with orthogonal polarization in the path between the Faraday rotator and servo AOM. Rayleigh backscattering strength lies in the range of a few ppm per meter [62]. This may result in a ghost-beam-induced phase error ϕerr=asp/asinϕsp [62] (a, amplitude of the signal; asp, ϕsp, ghost beam amplitude and phase, respectively). The induced fractional frequency uncertainty contribution depends on the dynamics of the phase of the ghost beam. In the worst case of a π-rotation of ϕsp over the measurement time t, the uncertainty contribution is ≤5×10−18/(t/s). A similar calculation for the estimated 40 dB power return loss of the AOM gives an uncertainty estimate of ≤1.7×10−17/(t/s). The second ghost beam source is light that is transmitted through the PR mirror and that is then reflected by the NBS. This ghost beam can make a second round through the IFL leading to a beat at the same round-trip frequency as the IL beat. From the spurious beats in the measured signals, we estimate spurious signal contribution of at least 30 dB below the IL stabilization beat leading to an estimated uncertainty contribution of ≤5×10−17/(t/s). These estimations show that, for the typical duration of the continuous measurement runs of ≥50ks, the estimated uncertainty contribution is on a relevant level of ≤1×10−21. In the future, the ghost beam uncertainty contribution can be further minimized by using, e.g., a PBS instead of an NBS as the element tapping of input light for the OOL measurement.

Finally, the uncharacterized OOL processes may result from electronic contributions, e.g., via the temperature dependence of the components. We estimated the electronic noise contribution of the control loop by deriving a surrogate for the photodiode signal from a commonly referenced synthesizer and connecting the voltage-controlled oscillator (VCO) output to the OOL signal tracking filter input [see Fig. 3(a)]. The gray curve in Fig. 6(a) shows that the instability of this signal is more than 1 order of magnitude below the observed OOL signal instabilities. Hence, electronic components within this loop can be ruled out as a possible cause, so that only processes in the photodetectors remain as the possible origin.

We have presented a compact, free-space interferometer layout for OOL characterization that uses an optically contacted, monolithic assembly for the reference arm rendering it robust against external disturbances and misalignment. Using this interferometer layout, we improve the OOL characterization resolution in fiber-based frequency transfer to ≤2.7×10−21. The temperature sensitivity of the OOL signal path is greater than that of the reference arm, so that the OOL signal quantifies the maximum remote place frequency transfer error induced by sender side temperature variations. Estimations of the impact of temperature and pressure variations based on the properties of the materials involved are close to the observed out-of-loop characterization signal variations. This opens up the possibility of post-correction for these error sources [30,37], while further studies have to elucidate the origin of the observed discrepancies below the 10−21 level.

In the current implementation the temperature sensitivity has been limiting for averaging times of 1000 s and longer. In the future, this constraint can be lowered further with refined layouts (see Appendix A) and active temperature stabilization.

We believe this research will stimulate further advances in lowering the uncertainty of optical frequency transfer and analysis of its fundamental limitations, and it will prove valuable for applications such as chronometric leveling between two non-laboratory sites.

Acknowledgment. We thank Alexandre Didier and Tanja Mehlstäubler for providing the UV curable adhesive, the group of Uwe Sterr and Tim Müller from the Department of Hydrology and River Basin Management of the Technical University Braunschweig for providing the air pressure measurements. We are grateful to Thomas Legero and Uwe Sterr for the PT100 temperature sensor calibration. We acknowledge fruitful discussions with Jochen Kronjäger, Erik Benkler, Alexander Kuhl, Thomas Waterholter, Andreas Koczwara (all PTB), Gudrun Wanner (AEI), Stephan Anders and Peter Zimmermann (both Layertec). The published work is part of the TiFOON project. This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. In addition, we received funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy Quantum Frontiers.