Laboratory Demonstration of the Local Oscillator Concept for the Event Horizon Imager

Black hole imaging challenges the 3rd generation space VLBI, the Very Long Baseline Interferometry, to operate on a 500 GHz band. The coherent integration time needed here is 450 s though the available space oscillators cannot offer more than 10 s. Self-calibration methods might solve this issue in an interferometer formed by 3 antenna/satellite systems, but the need for the 3rd satellite increases mission costs. A frequency transfer is of special interest to alleviate both performance and cost issues. A concept of 2-way optical frequency transfer is examined to investigate its suitability to enable space-to-space interferometry, in particular, to image the 'shadows' of black holes from space. The concept, promising on paper, has been demonstrated by tests. The laboratory test set-up is presented and the verification of the temporal stability using standard analysis tool as TimePod has been passed. The resulting Allan Deviation is dominated by the 1/$\tau$ phase noise trend since the frequency transfer timescale of interest is shorter than 0.2 s. This trend continues into longer integration times, as proven by the longest tests spanning over a few hours. The Allan Deviation between derived 103.2 GHz oscillators is $1.1\times10^{-14}/\tau$ within 10 ms<$\tau$<1,000 s that degrades twice towards the longest delay 0.2 s. The worst case satisfies the requirement with a margin of 11 times. The obtained coherence in the range of 0.997-0.9998 is beneficial for space VLBI at 557 GHz. The result is of special interest to future science missions for black hole imaging from space.


Introduction
Sagittarius A* is the handy object for the gravity theory choice, being a bright object of big angular size and well-known mass. An image with an angular resolution of 5 as is necessary for this test of competing theories. An imaging \microwave" band below 500 GHz is impossible because the interstellar clouds between us and Sagittarius A* broaden the instrument beam above the 5 as. The di®raction theory establishes the size of the maximum baseline to be 25,000 km i.e. more than twice bigger than the Event Horizon Telescope. The imaging from the ground is challenging on a twice higher frequency, if practically possible at all, because of tropospheric phase corruptions over the Earth-scale distances. Hence, the use of a space-based instrument is inevitable. Independent individual oscillators, even the best primary frequency standards, cannot assure coherent operation of the system. Jennison phase closure may solve the issue, though the minimum number of required satellites is 3, a notable cost increase over a system consisting of only 2 satellites. Hence, the frequency transfer between 2 satellites is of interest.
The Event Horizon Imager (EHI) mission concept (Roelofs et al., 2019;Kudriashov et al., 2021) (a) aims for an angular resolution of 5 as, opening up the possibility of testing theories of gravity, acquiring far more accurate measurements of both the black hole parameters (mass and spin), and deepening the understanding of both the behavior of the accreting plasma near the event horizon and direct manifestation of general relativity. Other science cases, including the imaging of water in planetforming disks (Gurvits, 2021;Haworth et al., 2019) might be also enabled by EHI.
The lowest-possible observation frequency which allows both mitigating the scattering by interstellar electrons in the Galactic plane (which would distort the image of the black hole in the center of our galaxy, Sagittarius A*) and imaging planet-forming disk, is 557 GHz. For the desired resolution of 5 as, the required distance between the radio-telescopes (i.e. the baseline of the interferometer) is more than twice the Earth's diameter. Moreover, atmospheric e®ects make observations from the ground at such submillimeter wavelength extremely challenging. For these reasons, a space-based array for Very Long Baseline Interferometry (VLBI) at 557 GHz is required to image Sagittarius A* and water disks at¯ne resolution.
With single-piece dish antennas limited to circa 4 m diameter to¯t into the launch vehicle, both the required integration time for detection and imaging time for obtaining high-quality images are challenging. The design, therefore, follows an alternative concept of interferometry using telescope-satellites in Polar Medium Earth Orbits involving inter-satellite optical links for local oscillator connection between the satellites, data transfer and metrology. This connection realizes the symmetric two-way frequency transfer concept as shown in Figs. 4 and 5 of Martin-Neira et al. (2019). The concept achieves coherent local oscillators at the two satellites and alleviates the need for the knowledge accuracy of the inter-satellite velocity for the transfer. A laboratory demonstrator has been built to put the concept to test using real hardware. This paper reports on the concept test results, including the high-level block diagram of the set-up and the measurement approach. The results conclude that this concept would be highly useful for a mission like the Event Horizon Imager.

State-of-the-Art Oscillators
A quantity of special interest in VLBI is the coherence time (Thompson et al., 2017, p. 434). The approximate coherence time (Rogers & Moran, 1981, Eq. (1)) is that time c for which the rms phase error is of 1 radian where f is the instrument center input frequency, is the integration time (in the white phase noise region) and AE ðÞ is the Allan standard deviation (or total two-point rms deviation) of two independent oscillators embarked on two separate telescope-satellites, with 1 ðÞ ¼ 2 ðÞ ¼ ðÞ being the Allan deviation (ADev) of each oscillator.
The longest integration time in EHI is to fall within about 70 s < < 450 s (Roelofs et al., 2019), as constrained by uv-smearing time. At the required frequency of f ¼ 557 GHz, the coherence time provided by the best H-maser for space use, the ACES-ESA active H-maser (AHM), is of only 10 s (Fig. 1). The coherence time provided by the iMaser-3000, a The \Event Horizon Imager" project of the Radboud RadioLab, https://www.ru.nl/astrophysics/radboud-radio-lab/ projects/ehi/. Fig. 1. Allan Deviation of oscillators.
the best-known ground H-maser, is not much longer, 40 s. Other potential future oscillators are not further addressed due to constraints in technology readiness level. Because the EHI is a future mission concept, the performance of the iMaser-3000, the best oscillator, is used as a reference for the coherence calculation. Over a short time period , the trend of ðÞ follows À1=2 and the squared product [ Â y ðÞ] 2 is << 1, hence (Rogers & Moran, 1981, Eq. (18)), the coherence WFN at ¼ 100 s is The coherence of 0.8 is not a®ordable because of the tight sensitivity budget (Kudriashov et al., 2021, Figs. 6 and 8), which on the other hand, assumes integration times of up to 491 s. Hence, the experimental demonstration of the devised frequency transfer concept is presented in this paper.

The Concept Under Test
The concept, described in Secs. 2.1 and 2.3 of Martin-Neira et al. (2019), aims at enabling VLBI between two or more satellites at sub-millimeter wavelength. All satellites need to operate coherently with the same Local Oscillator (LO). One-way frequency transfer (master-slave concept) does not work due to Doppler and gravitational redshifts. Instead, the concept under test relies on a two-way optical inter-satellite link and a mixing of the local and the remote local oscillators, and minimizes both the differential Doppler and gravitational redshifts.

Phase noise cancellation
The variance of the relative phase between the generated local oscillators at the two remote satellites is given by Martin-Neira et al.
where M and N are the frequency multiplication factors in a microwave set-up, is the integration time, is the integration variable (a time delay), g is the geometric delay of the incoming radiation from the target towards the two satellites, l is the inter-satellite link delay, and À 0 is the autocorrelation function of phase noise, i.e. the Fouriertransform of the phase noise power spectral density S ' ðfÞ ¼ 2LðfÞ in rad 2 /Hz. It is noted that the individual master clock phases do not appear in Eq. (4). Also, since the autocorrelation is an even function, the term (4À -À -À -À -ÀÞ vanishes at identical arguments in all À 0 functions, and the¯rst factor within the integral nulls at ¼ . This leads to variance, which is of 0.1 rad 2 at the worst sum-delay, as numerically computed in Fig. 2.

Concept description
The concept comprises two twin sets of microwave blocks assumed to be on board two satellites (Fig. 3). Oven Controlled Crystal Oscillators (OCXO) are used as master oscillators, at both satellites. The OCXO signal (e.g. 100 MHz) gets Â M frequency multiplied and sent to the other satellite via a dedicated optical inter-satellite link at point A (B). This tone is received at point A′ (B′), slightly o®set by a Doppler shift, hence the superscript prime. The concept aims at producing equal sum frequencies at the output of the mixers by adding the frequencies at points A and B′ (B and A′). The mixers are con¯gured as up-converters with both IF and LO inputs having about the same frequency and the output having the sum frequency, that is, IF = LO = RF/2. At the output of the mixers, some harmonics are found, like those at a frequency of 2 Â LO and 2 Â IF, due to the limited rejection of the mixer, bringing the (interfering) individual oscillator phase to RF, which is an unwanted contribution to the concept (4). Hence, the importance that the double-balanced mixer suppresses very well the even harmonics. The mixer RF output frequency is then OCXO Â M Â 2, and is further ÂN frequency multiplied, providing an LO frequency output to the science instrument of OCXO Â M Â 2 Â N. The frequency di®erence between the outputs onboard the two satellites LO1-LO2 is desired to be null.

High-level block diagram
The high-level block diagram of the concept described above is shown in Fig. 4. A large M/N ratio between the frequency multiplication factors is desired to mitigate the potential e®ect of the temperature di®erence between the EHI satellites. When analyzing in detail, the value of the M/N ratio is seen to be constrained by the highest frequency that the optics can operate. The proximity to RF of both the 2 Â IF and 2 Â LO harmonics requires the use of a double-balanced mixer, a Hartley mixer with 180 phase shifts or a Gilbert cell, while a sub-harmonic mixer brings unwanted asymmetry and complexity.
Marki is the only known supplier of balanced mixers above 50 GHz (the highest RF is of 80 GHz) ful¯lling the LO = IF = RF/2 condition. The highestfrequency IQ-mixer operates at 110 GHz and hence, the di®erence between derived LOs can hardly be measured above 110 GHz unless VNA extenders are used as frequency down-converters. Space heritage on multipliers and ampli¯ers runs thin above 100 GHz. To make the hardware compatible with a possible follow-on demonstration at a highfrequency radio telescope, the output frequency and power level have to be within 93-121 GHz and 13-18 dBm, respectively (Mattiocco et al., 2015). This relaxes the output frequency by a factor 5 when compared to EHI though medium power ampli¯ers are still required at 100 GHz outputs.
It is assumed that: (a) a 100 MHz OCXO is preferred over either 5 or 10 MHz OCXO followed by a chain of frequency multipliers and ampli¯ers; (b) a PLDRO with the highest-possible output frequency is to be selected to minimize the number of frequency multipliers and ampli¯ers to achieve the Â M factor, hence the use of a PLDRO at 8.6 GHz; (c) the optoelectronics can operate at a frequency up to 50 GHz; and (d) a bu®er ampli¯er is needed at the photodetector output to supply enough IF power to the mixer. The active frequency doubler at the RF output of the mixer has been assembled using COTS ampli¯ers and a frequency doubler mounted in cavities inside a box. The list of all  Table 1.

Measurement Approach
The highest frequency of the Miles TimePod 5330 equipment is 30 MHz, much smaller than the output frequency of 100 GHz of the breadboard and hence, another approach is needed to test the concept. A VNA-based measurement approach has been chosen. The tones under test are input to two (or four) VNA channels. Internally, the VNA receivers downconvert the inputs to an IF band of up to 15 MHz width, sampled at 100 MS/s. The VNA then  The set-up includes also Keysight N5227B PNA Microwave Network Analyzer, OML WR-06 extenders driven by Agilent PSG E8267D generator, two spectrum analyzers, and 10 MHz discipline for them. These units are used to measure the performance, they are not part of the breadboard.
provides the (unwrapped) phases of incoming signals with respect to the time, at a set-up (nominal) frequency. The measurements are stored in memory and post-processed using Excel and Stable32. Excel is used to calculate both the VNA sampling time and the phase di®erence. Stable32 has been employed to obtain the Allan Deviation. Calculations have been con¯rmed using the known equations presented in the following section.

Computing Allan Deviation from VNA measurements
The time error is where 'ð; tÞ is the phase di®erence between the input tones to the VNA, ¼ B À1 IF ¼ tði þ 1Þ À tðiÞ is the sampling time interval (inverse to the set-up VNA IF bandwidth, as corrected for the VNA dig-ital¯lter and recording length), and f 0 is the set-up VNA frequency. The phase di®erence is free from the internal phase in each of the VNA channels because they are all driven by the same oscillator, that is 'ð;tÞ¼ ' 1 ð;tÞÀ' VNA ð;tÞÀ½' 2 ð;tÞÀ' VNA ð;tÞ where ' 1;2 ð; tÞ are the measured phases in the VNA channels and ' VNA ð; tÞ the intrinsic channel phase. The VNA recording is exported to Excel to calculate both the sampling time and the time error x k ð; tÞ in Eq. (5). This¯rst di®erence x k ð; tÞ has the dimension of time and is a function of time. It can be understood as the phase di®erence, in time units, between two inputs accumulated over a time interval . The corresponding fractional frequency is This second di®erence is taken between two consecutive samples x k ð; tÞ and x k ð; t À Þ. The fractional frequency y k ð; tÞ is a dimensionless function of time t, at observation duration . It has the sense of a fractional frequency ðf À f 0 Þ=f 0 . The ADev can be built from Eq. (7) as a third di®erence where the brackets h i denote averaging, and factor 0.5 stands because the phase noise LðfÞ is de¯ned as where the phase instability S ' ðfÞ is a one-sided function that represents the two-sided power spectral density of the phase°uctuation, as in Eq. (4). Averaging N non-overlapping numbers of y k ð; tÞ allows studying y ðNÞ. The value of N should exceed one-tenth of the data set duration as otherwise, the number of third di®erences which can be built is too few to provide reliable statistical results. The measurement approach was validated with Miles TimePod.

Optoelectronics
As shown in Fig. 5, two LO signals are exchanged over a 5 km long¯ber wound in a reel using two  wavelengths. The local and the received remote signals (converted to electrical) are then fed to the VNA, phases are sampled and stored in a¯le. O®line in Excel the phases are added to mimic the two derived local oscillators at the sum frequency in Fig. 4, and the di®erence of the generated sums is the¯nal output, implementing the functionality of the IQ-mixer. The measurement approach described in the former section was followed.
The optoelectronics (Fig. 5(a)) can operate above their nominal highest frequency of 18 GHz of its modulators (iXblue MXER-LN-20). Local Oscillators have been realized at 36, 51.6 (our baseline frequency) and 88 GHz. The ADev between the local oscillators is compliant with the requirement (Fig. 5(b)). The phase coherence was maintained without showing any drift or variation beyond thermal noise, hence indicating such coherence could have been extended for a considerably longer time interval.
The y ðÞ < 3 Â 10 À14 / trend in the Allan Deviations was achieved using available equipment and components in the lab. Small di®erences between the Allan Deviations at these Local Oscillator frequencies may be driven by the use of di®erent cables, splitters, and also by the phase noise di®erence between two types of lab generators (Agilent E8257D with low noise option and E8267D at 18 GHz and 25.8 GHz, respectively).

Balanced mixers
The balanced mixers are introduced in Fig. 6(a). They are mixing the local and remote local oscillator signals. The frequency o®set between the LO (local LO) and the IF (remote LO) input tones to the balanced mixer a®ects the Allan Deviation. At Á ¼ 0, the balanced mixers Marki MM1-2567L o®er a 2Â LO to RF isolation >30 dB (including the mixer loss). The result in Fig. 6(b) shows that the coherence requirement for the desired 557 GHz channel observations is ful¯lled. This was achieved by combining two independent 24.5 GHz laboratory local oscillators with a separation of Á ¼ 5 MHz into a¯nal one beating at the sum frequency, i.e. circa 49 GHz. The plot below also shows that the EHI local oscillator concept surpasses the performance of the state-of-the-art iMaser-3000 oscillator, enabling much longer integration times as required in the space-to-space VLBI.

E®ect of mixer input frequency o®set on performance
The science interferometer maximum frequency can be approximated as (Eqs. (1) and (2)) This equation can be used along with the ADev ðÞ ¼ 1 (i.e. 1/Þ trend in the white phase noise dominated region. The result in Fig. 7 indicates that an f max ¼ 557 GHz is achievable with a frequency  o®set of Á ¼ 1 kHz. The di®erence between the performance at 5 MHz frequency o®set presented in Fig. 6(b) and the one shown in Fig. 7 is attributed to the re-assembling of the set-up using similar components after 3 months.
In the particular application of EHI, the intersatellite link Doppler shift is proportional to the frequency o®set between the frequencies exchanged. This Doppler shift can be neglected only when the frequency o®set is below 1.1 kHz. There are several EHI con¯gurations possible. In one of them, the satellites have a 1000 km height o®set rather than only 23 km, and for this one, the Doppler shift over the inter-satellite link is larger, requiring the frequency o®set to be below 27 Hz before a correction is needed. However, Fig. 7 shows that the crossing point between the measured f max and the 557 GHz requirement occurs for a frequency o®set of 1 kHz, and a custom mixer connection permits Á of 50 Hz (Sec. 5.6). It is then concluded that a correction for the Doppler residual e®ects is very likely to be required in EHI with a big altitude o®set of 1000 km.

OCXOs and PLDROs
As another step in the concept demonstration, the laboratory frequency generators have been replaced by master OCXOs and PLDROs (Fig. 8(a)). Because of the di®erence in the phase noise between the lab and the set-up generators (OCXOs followed by PLDROs), a new slightly higher frequency o®set crossing point is found within 1 < Á < 1:2 kHz ( Fig. 8(b)). The resulting Allan Deviation in Fig. 9 con¯rms that the coherence requirement for the desired 557 GHz channel observations is ful¯lled at Á ¼ 1:36 kHz.
Concerning the frequency o®set Á, OCXOs frequency drift due to aging and radiation e®ects shall be accounted for over a mission lifetime of 5 years. For this class of OCXO the frequency drift is expected to be in the order of 140 Hz. The impact of radiation e®ects is 0.15 Hz, as simulated in SHIELDOSE via SPENVIS, within a worst-case scenario. Being the frequency drift, in the¯rst approximation, a deterministic process linear in time, the maximum overall frequency di®erence between the two OCXOs is 280 Hz. This translates to 73.2 kHz out of the balanced mixer due to frequency multiplication from OCXO to the optical ISL. In case a narrower frequency di®erence is needed, the instant frequency di®erence can be measured and tuned-out by OCXO(s) control voltage. This can be implemented by measuring the \minus" output of the balanced mixer and using this error signal within a feedback loop.

K-band active frequency tripler and bu®er ampli¯er
With the addition of two sets of K-band Active Frequency Triplers and Bu®er Ampli¯ers to the master OCXOs, PLDROs and mixers, the local oscillator demonstrator setup consists now of building  (panel (a)), and f max with respect to the Á (panel (b)). blocks exclusively (Fig. 10(a)). The measured ADev between the derived 51.6 GHz LOs allows for a f max ¼ 10:6 THz (11), at an arbitrary K-band frequency o®set of Á ¼ 63:2 kHz. As shown in the Fig. 10(b) the coherence requirement for the desired 557 GHz channel observations is ful¯lled with an ample average margin of 20 times.

E®ect of mixer connection arrangement on performance
There are di®erent possibilities of connecting the local and remote oscillator signals to the mixer to output the desired¯nal local oscillator. The mixer has three ports, all of them which can be used either as input or output. It was found that by selecting the RF and LO ports of the mixer as inputs and the IF as output (labeled as RF À LO = IF), the frequency o®set Á at which the 557 GHz requirement was ful¯lled improved (decreased) by a factor 10 ( Fig. 11). Moreover, such narrowing was not dependent on the VNA IF bandwidth. Hence, this connection arrangement of the 51.6 GHz balanced mixers would allow keeping the frequency separation between the 25.8 GHz local oscillator components within of 1 kHz, avoiding the need for a Doppler correction, at one of the operation modes. However, to¯lter out the harmonics which appear next to the wanted sum-frequency at the output of the mixer, it is impractical to have a too narrow frequency o®set to start with.

The breadboard
The standard mixer connection is used to keep the nominal output power. An active frequency doubler (Table 1) is temporarily used without the MPA, at branch 1 (green at Fig. 12).   The 103.2 GHz derived oscillators are beyond a VNA operation frequency band (Sec. 3). These oscillators are input to VNA frequency extenders (Table 1). Lab generator drives these extenders at 10.35 GHz. The receiving part of extenders forms 10:35 Â 10 ¼ 103:5 GHz local oscillators. Extenders perform frequency down-conversion (from 103.2 GHz to 300 MHz). The output from extenders is within the VNA operation frequency band. This permits to use of the measurement approach (Sec. 3).
The function f max ðÁÞ at 103.2 GHz ( Fig. 13(a)) has three distinguishable regions namely, the smallest Á up to 15 kHz featuring many interfering peaks at the worst S/N near 1 kHz, the 15 kHz < Á < 30 kHz featuring a very good performance increase, and the best S/N range Á > 30 kHz. The measurement duration at Fig. 13(a) is 90 s that constraints the maximum ADev by about 18 s (by default set-up at the Sta-ble32 program), may hide a long-term degradation of ADev and°atten slightly the latter region. The former region is also about twice better than Fig. 11(a) due to tuning of power over optical¯ber, and the use of isolators suppressing leakages through mixers.
The ADev 1:1 Â 10 À14 / between 103.2 GHz oscillators has been measured at Á ¼ 210 kHz over 1.75 h (Fig. 13(b)) while the phase di®erence between these 103.2 GHz oscillators does not demonstrate any ramp over 4 h (we did not test further). The performance decay towards 1000 s vanishes at longer measurement time of 2 h. This ADev degrades 2.15 times at the maximum EHI system delay (this is the maximum sum of communication and geometric delay, see Table 2). The average margin between the degraded performance and the requirement is 11.2 times. Tests showed the same performance at (case 1) two branches at similar temperature, (case 2) two branches at the same temperature 22 C and (case 3) two branches at the temperature di®erence 30 À 20 ¼ 10 C constant over measurement time (2-3 h). We do not know any other oscillator satisfying this requirement (Fig. 1). Hence, this is a game-changer for the third generation space Very Long Baseline Interferometry.
The interferometer coherence is one of the factors in the sensitivity calculation. The target is to assure the coherence of 0.85 (Rogers & Moran, 1981), at white phase noise. The obtained coherence varies with respect to the correlator integration time (0.1 s < < 450 s) in range of 0.9992À0.9998, at 557 GHz (Rogers & Moran, 1981, Eq. (14)). Because the ADev degrades twice (Table 2) at the longest baseline 25,000 km, the corresponding coherence degrades to the range of 0.997-0.9992. The ratio of the obtained coherence to a nominal coherence 0.85 (Rogers & Moran, 1981) is in range of 1.17-1.18 that is useful for VLBI instrument sensitivity. A similar issue is discussed in (Kudryashov, 2018).
Narrowing the VNA IF bandwidth below 100 Hz improves the ADev (by¯ltering-out unwanted interferences) indicating that the performance \ceiling" has not been achieved yet. A future improvement, if any, may leverage on analysis of phase noise between these 103.2 GHz oscillators and peaks therein.

E®ect of the Inter-Satellite Link Delay
The performance degradation with respect to the inter-satellite link delay has been studied by introducing an arti¯cial delay in post-processing of the VNA recordings.
The block diagram in Fig. 14(a) has been used, which involves two free-running lab generators and the VNA. Di®erently to Fig. 6, the inter-satellite link delays are added arti¯cially to the VNA recordings. The longest-possible delays (altogether inter-satellite link delay and science signal geometric delay) are 0.2 s in EHI and 16.7 s (5,000,000 km) in ESA's LISA project. Our local oscillator concept has been tested over delays of up to 20 s. It has been observed that the relative phase between the two derived LOs grows about linearly with the arti¯cial delay ( Fig. 14(b)) at a rate of the order of mHz. There is no frequency discipline between the lab generators used in the experiment. Furthermore, it has been veri¯ed that the concept still works well even when there is a large delay asymmetry (between the two directions of the inter-satellite link) of up to 60 m. Using the same technique, the impact of the inter-satellite link delay on the standard deviation of the phase noise has also been assessed using the set-up shown in Fig. 13(a) but with the lab generators disciplined on two iMaser-3000 from the ESTEC UTC lab. The measurements are given in Table 2. The standard deviation of the phase grew only by a factor 7 over six orders of magnitude increase in the inter-satellite link delay, from 16.7 s to 16.7 s. This phase change is far smaller than the phase change of each generator.
In the case of the longest baseline of EHI the inter-satellite link delay is of 180 ms and the ADev increases a factor of 2.15 and the phase ramp is smaller than 0.3 mHz.

The Breadboard Modi¯cation for Future
Operation Onsite IRAM Tests onsite IRAM are intended to follow-on the lab demonstration. These tests may consist of a use of two derived oscillators 103.2 GHz for band 4 receivers at the PdBI and a performance comparison to the nominal operation mode, at the constant interferometer baseline. One must feed-in the LO/3 frequency within the frequency range of 93-121 GHz and power level range of 13-18 dBm through WR-10 waveguide. Hence, there is no need to keep the Á within 1 kHz, at the next stage. Because the IRAM band 4 is 283-365 GHz, its coherence requirement is mitigated compared to EHI.
A¯lter at the mixer output should pass the derived sum-frequency RF = IF + LO and reject altogether tones 2 Â LO, 2 Â IF, 2 Â LO -Á, etc. The frequency separation between the RF and either 2 Â LO or 2 Â IF is the Á. Notch¯lter su®ers from the fact that derived sum frequencies drift (700 Hz over 2 h warm-up) and hence, a bandpass lter is required. The fractional bandwidth of a practical bandpass¯lter at 51.6 GHz is 3% ($1.6 GHz) while 1.5 % is very challenging.
OCXOs and PLDROs permit Á up to 256 kHz and 680 MHz, respectively. Main RF blocks of the breadboard permit Á up to 2.6 GHz satisfying the desired fractional bandwidth of 3% with a margin however, isolators available temporarily shrink the Á to 100 MHz. Hence, breadboard modi¯cation for use onsite IRAM consists of use of bigger Á leveraging on lab generator(s) 8.6 GHz.

Conclusions
In this paper, we examined the concept for the local oscillators' connection for the Event Horizon Imager (EHI) mission concept.
The Allan Deviation 1:1 Â 10 À14 / of the local oscillator demonstrator to perform space-to-space Very Long Baseline Interferometry for the Event Horizon Imager mission concept has been measured using exclusively building components successfully, at 103.2 GHz within 10 ms < < 1000 s. Cross-check with TimePod veri¯ed the measurement approach. The demonstrator con¯rms the concept under test. The breadboard result shows also that the coherence requirement for the desired 557 GHz channel observations is ful¯lled with an ample margin 11 times. This factor takes into account the performance degradation at the longest communication and geometric sum delay. The instrument sensitivity gets a useful margin of 17-18%.
More in detail, the optoelectronic part has been tested for phase noise, Allan Deviation, and phase drift using a 5 km optical¯ber length, to produce 36, 51.6, and 88 GHz LOs. The optoelectronics allows ful¯lling the required Allan Deviation with over Â25 margin. The OCXOs and PLDROs permit a f max ¼ 557 GHz, at a frequency o®set of Á ¼ 1-1.2 kHz on 8.6 GHz inputs. Both K-band active triplers and bu®ers demonstrated a f max ¼ 10:6 THz, at an arbitrary K-band frequency o®set of Á ¼ 63:2 kHz. With a custom mixer connection arrangement this o®set Á could be made far narrower, even below 1.1 kHz, as required to avoid Doppler-correction at the nominal operation mode for imaging Sagittarius A*, M87*.
The quality decay is only a factor of two, at the longest delay. The concept operates at a large delay asymmetry of 60 m and LISA-ESA scale intersatellite link delays.
There is no performance degradation due to the breadboard modi¯cation as required to operate onsite IRAM namely, both results obtained with lab generator satisfy the coherence requirement of IRAM band 4 (and also the EHI), including the case when the frequency separation of the signals exchanged over the optical¯ber is Á ¼ 100 MHz.
The paper provides the current progress on the critical element of the Event Horizon Imagerthe local oscillator breadboard. The paper also indicates the foreseen avenue for further investigation on LO sharing scheme.