Ultra stable and very low noise signal source using a cryocooled sapphire oscillator for VLBI
Abstract
Here we present the design and implementation of a novel frequency synthesizer based on low phase noise digital dividers and a direct digital synthesizer. The synthesis produces two low noise accurate and tunable signals at 10 MHz and 100 MHz. We report on the measured residual phase noise and frequency stability of the synthesizer, and estimate the total frequency stability, which can be expected from the synthesizer seeded with a signal near 11.2 GHz from an ultrastable cryocooled sapphire oscillator.
The synthesizer residual single sideband phase noise, at 1 Hz offset, on 10 MHz and 100 MHz signals, respectively, were measured to be 135 dBc/Hz and 130 dBc/Hz. Their intrinsic frequency stability contributions, on the 10 MHz and 100 MHz signals, respectively, were measured as and , at 1 s integration time.
The Allan Deviation of the total fractional frequency noise on the 10 MHz and 100 MHz signals derived from the synthesizer with the cryocooled sapphire oscillator, may be estimated as and , respectively, for 1 s s.
We also calculate the coherence function, (a figure of merit in VLBI) for observation frequencies of 100 GHz, 230 GHz and 345 GHz, when using the cryocooled sapphire oscillator and an hydrogen maser. The results show that the cryocooled sapphire oscillator offers a significant advantage at frequencies above 100 GHz.
phase noise, frequency stability, cryogenic sapphire oscillator, frequency synthesizer, VLBI coherence
1 Introduction
\IEEEPARstartVerylongbaselineinterferometry (VLBI) radio astronomy requires a lowphasenoise ultrastable frequency reference. Such a reference is commonly derived at 10 MHz from a hydrogen maser. The development of a cryocooled sapphire oscillator (cryoCSO) and frequency synthesizer described here offers a lower noise alternative.
Cryogenic sapphire oscillators (CSO) [1, 2, 3, 4, 5, 6, 7] have demonstrated extremely good frequency stability (characterized in terms of Allan deviation) at integration times in the range 1 s to 1000 s. However, these oscillators do not operate at precisely prescribed repeatable frequencies. Transferring their stability to precisely 10 MHz and 100 MHz presents technical challenges [8, 9, 10, 11].
In this paper, we describe the design and implementation of a synthesizer based on an ultrastable CSO [12, 4, 5] complemented by very low phase noise digital dividers and a direct digital synthesizer (DDS) [13]. We evaluate the performance of the synthesizer in terms of phase noise and stability of the synthesized 10 MHz and 100 MHz reference signals.
2 Frequency synthesizer design and measurement techniques
2.1 Synthesizer design and architecture
We constructed two nominally identical frequency synthesizers. Fig. 1 shows a block diagram of the synthesizer based on a stable microwave oscillator, the cryoCSO, and a highresolution DDS phase referenced to a subharmonic of the cryoCSO signal. The output signal frequency of the cryoCSO is GHz. This signal is not tunable and the closest integer multiple of 100 MHz is 11.2 GHz. By frequency shifting the cryoCSO signal with a XXXXX MHz signal, generated by the DDS unit, this produces the correct frequency for integer digital frequency division. The frequency shifting was performed with an image rejection mixer in order to suppress the spurious mixing product about 4 MHz above the useful signal. Without this the spurious signal would impair the divider operation and filtering it out would require the use of an impractically highQfactor microwave filter. The use of the image rejection mixer is an elegant alternative to solve the problem.
By combining an Xband IQ commercial mixer with a custom made /2 phase shifter we realized an image rejection of about 40 dB. The signal levels are optimized to suppress the carrier by about 40 dB. At the output of the image rejection mixer a 2step digital frequency division by 56 produces a 200 MHz output. This 200 MHz signal, downconverted from the cryoCSO, is used to clock the DDS synthesizer, as well as a reference signal for the phaselocked loop controlling the frequency of a low phase noise 100 MHz quartz oscillator, which is frequency doubled by a commercial low phase noise frequency multiplier. The phaselocked loop ensures that the ultrahigh frequency stability of the cryoCSO is transferred to the 100 MHz quartz oscillator with only a small addition of noise. The frequency of the latter is further divided to 10 MHz, but with a nonnegligible noise contribution from the divider.
Furthermore the circuit provides a 1.4 GHz by dividing the output signal of the mixer by 8. This signal, which is very close to the hydrogen maser microwave transition, is a good candidate for future very low noise synthesizers. It is worth noting that the technique described here can easily be adapted to any of the possible microwave output frequencies from the cryogenic sapphire oscillator.
Component  Output freq.  @  

Model Number  [dBc/Hz]  MHz  100 MHz 
HMCC007 ( 8)  120  1,400  
HMC365S8G ( 4)  2,800  
HMC705LP4 ( 14)  124  200  130 
HX4210( 10)  135  10  
ERA5+ amp  120  2  161 
ZFL500LN+ amp  144  200  150 
AD9912 DDS  103  2  144 
50104517D quartz  70  100  134 
cryoCSO  97  11,202  138 
Not measured individually, only used as input to HMC705LP4.
Only individually measured at 100 MHz, where its phase noise could not be measured above the measurement noise floor.
The free running phase noise for the oscillator at 100 Hz offset is 130 dBc/Hz [14] and with a dependence = 70 dBc/Hz at 1 Hz. In the last column the residual phase noise inside the PLL is shown where it was measured using two oscillators locked to a common 100 MHz signal with a locking bandwidth of about 100 Hz.
Absolute phase noise from [5]
We are interested in their contribution on the 100 MHz carrier at the output of the synthesizer.
For the ERA5+ amp, the DDS and the cryoCSO division by 56 times means 35 dB reduction to their contribution at 200 MHz and 41 dB at 100 MHz.
2.2 Measurement techniques
Phase noise measurements are traditionally made using the analog technique by mixing equal frequency signals 90 out of phase as shown in Fig. 2(a). The baseband signal voltage is then sampled and Fourier analyzed by a FFT spectrum analyzer, and then postprocessed to extract the phase noise data.
Initially, we used the analog technique to evaluate the residual phase noise of the components used to realize the synthesizer. See Fig. 3. In a second phase we used a Symmetricom 5125A phase noise test set; hereafter referred to as the test set (see Fig. 2). The test set outputs the single sideband (SSB) phase noise and the signal frequency stability of the device under test (DUT) in realtime by comparing it with a reference signal. The test set performs phase detection by digital signal processing (DSP) methods, sampling the RF waveforms directly [15].
3 Results and discussion
Oscillator phase noise can be described by a power law given by
(1) 
where is the power spectral density of phase fluctuations and has units rad/Hz. In the literature, phase noise is commonly reported as
(2) 
which has units dBc/Hz. In this paper all phase noise measurements are reported in or SSB units. On a loglog plot, the term maps on to a straight line of slope i 10 dB/decade. The value of the slope is used to identify the various noise processes commonly encountered in oscillators [16].
3.1 Residual phase noise and stability
Residual phase noise
We measured the residual phase noise of the DDS, dividers, and amplifiers (as the DUT) using setup Fig. 2(a) and show the residual phase noise of the DDS and dividers in Fig. 3. Table 1 summarizes the relevant phase noise contributions of the active components used in the design of the synthesizer. The power spectral density of phase fluctuations on the DDS 2 MHz output signal is 103 dBc/Hz at 1 Hz offset. Since these fluctuations are superimposed on the microwave carrier (when DDS signal is mixed with that of the cryoCSO) their power spectral density is reduced to 144 dBc/Hz when microwave frequency is divided to 100 MHz and is negligible in the total noise budget of the synthesizer. Its contribution is less than the phase noise of the cryoCSO signal referred to 100 MHz, which is 138 dBc/Hz at 1 Hz offset [5]. The last column of Table 1 lists the relevant noise contributions of each component referred to 100 MHz.
Using the cryoCSO output signal to drive the inputs of two nominally identical synthesizers we separately measured the relative phase noise (see Fig 2(b)) of the 10 MHz and 100 MHz output signals. The results are shown in Fig. 4 in curves (1) and (2), respectively, and specify the residual phase noise of a single synthesizer. Curve (3) is the phase noise of a single 100 MHz10 MHz digital frequency divider (model Holzworth HX4210, curve (4) from Fig. 3) used here. Curves (4) and (5) are the measurement noise floors for the test set at 10 MHz and 100 MHz, respectively. Curve (6) indicates the expected level of phase noise from the cryoCSO signal when divided down to 100 MHz. It contributes negligibly to the phase noise of the synthesizer at 100 MHz. The indicated spurs at multiples of the 1.46 Hz of the cryocooler compressor cycle result from poor rejection by the measurement equipment. This was proven when the cryocooler compressor, used to cool the resonator in our loop oscillator, was switched off yet another was still running nearby and the same spurs were observed. Note the power levels of these spurs are about 20 dB lower on the 10 MHz signal than on the 100 MHz signal. The spur near 60 kHz is the residual Pound modulation sideband from the cryoCSO loop oscillator.
From Fig. 4, one can clearly see that the bandwidth of the PLL controlling the 100 MHz quartz oscillator is about 100 Hz. At Fourier frequencies Hz the phase noise of the 100 MHz output signal (curve (2)) is due to the freerunning quartz oscillator (which is consistent with its specifications). At Hz, i.e. well within the PLL bandwidth, the phase noise of the 100 MHz output signal is due to intrinsic fluctuations of the RF mixer used in the PLL. In addition to the PLL mixer, the low frequency noise is also due to the intrinsic fluctuations in the frequency divider (division by 56) and in the 200 MHz amplifier. These are sources of uncorrelated phase fluctuations in two synthesizers. See Table 1 for their contributions on the 100 MHz carrier.
The residual phase noise of the 10 MHz signal (curve (1)) is clearly limited by the frequency divider phase noise at Fourier frequencies Hz Hz. The 10 MHz synthesizer phase noise is flicker phase dominated, and at Fourier frequencies Hz it exhibits flicker frequency noise. For Hz the shape of the the phase noise spectrum of the 10 MHz signal can be explained by the residual phase noise of the 100 MHz output reduced by 20 dB (the relative frequency reduction) plus the intrinsic noise of the divider, but for Hz it is not clear why we see some excess noise in the 100 MHz10 MHz frequency divider. We are currently investigating the effect of changing the operating temperature setpoint of the dividers and whether we can explain this.
From these data the SSB residual phase noise at 1 Hz offset on the 10 MHz and 100 MHz signals is 135 dBc/Hz and 130 dBc/Hz, respectively. The former is confirmed by the phase noise measurement of a single divider as indicated by (4) in Fig. 3. A very good 10 MHz quartz oscillator phase noise is 122 dBc/Hz at 1 Hz offset [20]. Our result here is 13 dB lower than the best quartz.
Frequency stability
Using the setup of Fig. 2(b), we measured the intrinsic frequency stability of the synthesized 10 MHz and 100 MHz references (curves (1) and (2) of Fig. 5). The measurements obtained with the test set that are shown in Fig. 5 are for a Nyquist equivalent noise bandwidth (NEQ BW) of 0.5 Hz. The NEQ BW determines the minimum integration or averaging time () for stability measurements and is given by . (See Ref. [15] for details.) The test set tabulates the Allan deviation of the fractional frequency fluctuations at certain integer multiples of . These data are used in the figures herein.
Curves (3) and (4) are the measurement noise floors of the test set, measured by splitting the outputs of the synthesizer with a power divider. These were then subtracted from curves (2) and (3) and the intrinsic stability () contribution for a single synthesizer at 10 MHz and 100 MHz calculated. At 1 s of integration, these are and , for the 10 MHz and 100 MHz outputs, respectively. The degradation in stability on the 10 MHz signal can be attributed to the phase noise contribution of the 100 MHz10 MHz divider itself (see Fig. 4). If the division to 10 MHz was noiseless we would expect the same fractional frequency stability on that signal as on the input 100 MHz signal.
It should be noted that the 100 MHz to 10 MHz frequency divider is temperature sensitive and the intrinsic stability shown in Fig. 5 is the best result obtained while actively controlling the temperature of the divider with a thermistor and resistive patch heater a few degrees above the ambient temperature.
Using the setup of Fig. 2(c) we measured the stability of the cryoCSO/synthesizer against our hydrogen maser at 100 MHz. The resulting Allan deviation of the combined fractional frequency fluctuations is given by curve (1) in Fig. 6. It is the best result we measured over a two week period. The hydrogen maser noise dominates the measurements out to about s of averaging. The circled data indicate the cryoCSO frequency stability where it is assumed that the cryoCSO dominates over that of the maser. To fully characterize the frequency instability of the synthesized signals for s a second cryocooled oscillator is needed.
From the previously measured frequency stability of the cryoCSO against a liquid helium cooled CSO [5] (curve (2) in Fig. 6) it is possible to estimate the expected short term and very long term stability of a single cryoCSO (curve (3) in Fig. 6). This estimate is based on the assumption that for times s both oscillators are equal in performance and at times s the frequency stability of the cryoCSO is that of curve (1), from the comparison of the cryoCSO with our maser. This must be the case else we would see the long term stability (no frequency drift removed) similar to that of curve (2) for s. For s the liquid helium cooled CSO frequency stability is worse than that of the cryoCSO. Hence we are able to calculate the total noise contribution from the cryoCSO and a single synthesizer for integration times s, where we have reliable data.
In order to calculate the expected stability of the synthesized signals with the cryoCSO the frequency stability of the cryoCSO (curve (3) of Fig. 6) was added to the intrinsic stability of the synthesizer (curves (1) and (2) of Fig. 5) at 10 MHz and 100 MHz, respectively. The results are shown in Fig. 7 where they are compared with the frequency stability of our maser (curve (1) for s) and with that of a high performance maser (curve (4)). Curves (2) and (3) are the frequency stabilities for the 10 MHz and 100 MHz cryoCSO/synthesizers, respectively.
This calculated data was interpolated to the measured comparison data between the hydrogen maser and the cryoCSO/synthesizer (curve (1) in both Figs 6 and 7) for where the noise contribution is assumed to be largely due to the cryoCSO. As a result we were able to curve fit to the data of curves (2) and (3), in Fig. 7, resulting in flicker floors estimated to be about (10 MHz) and (100 MHz) at integration times around 1000 s. These flicker floors cannot be well specified due to insufficient data.
However, since we only have one cryocooled sapphire oscillator we were not able to directly measure its stability for s s. A second cryocooled oscillator is necessary to do this. The fits (curves (2) and (3) in Fig. 7) represent the expected, yet optimistic, total stability of the synthesizer using the cryoCSO and may be described by
(3) 
and
(4) 
where the subscripts represent the particular output frequency of the synthesizer in MHz. These fits are valid for s.
The long term frequency stability at both 10 MHz and 100 MHz signals should converge due to the long term stability of the cryoCSO. However beyond s the cryoCSO stability is largely determined by the environment; temperature and pressure changes. Also there is an uncertainty about the frequency stability of the hydrogen maser, so the only way to truly establish this performance is with at least one more cryocooled sapphire oscillator.
3.2 Long term performance
The cryoCSO signal frequency is subject to drift due to ambient pressure and temperature changes in the lab, though the lab temperature is stable to C. These changes translate into temperature and pressure changes within the cryostat. The design of the cryostat has a mixed liquidheliumgas space. For a schematic of the cryostat see Fig. 1 of Ref. [5]. This means the region where the cryocooler condenser constantly reliquefies a small quantity of helium gas in a closed system. It has been found that best operating condition in this helium gas space is where the pressure is maintained as low as possible.
Figure 7 (curve (1)) shows a comparison at 100 MHz of the cryoCSO/synthesizer compared to our hydrogen maser. This data was taken over a period of about 2 weeks after the cryoCSO had been continuously operating for about 9 months. It has been observed that the data exhibit a long term trend of decreasing frequency. This measurement represents the quietest data segment in terms of low frequency drift that we have taken to date. The solid line (labeled (5)) in Fig. 6 is a fit used to determine the drift rate of /day assuming drift dominates over random walk of frequency. As the cryoCSO was improved on, a 4 K radiation shield reduced thermal gradients [5] and reduced frequency drift but the origin of the remaining frequency drift is unknown.
The normal operating pressure inside the helium gas space is about 46 kPa. A steady state of helium gas being reliquefied maintains this. If power is shut off to the compressor, for example, in the event of a power failure, the pressure in this region will begin to rise. This was observed, when, due to a maintenance issue, the power to the whole lab was shut off for about 20 minutes. After the power was restored the pressure stabilized to its normal condition within an hour and the oscillator started automatically and remained functioning.
Subsequently we measured the time evolution of the cryoCSO frequency after this event. This was measured by downconverting the 11.202 GHz signal of the cryoCSO to approximately 2 MHz with a 11.200 GHz signal produced from a higher order harmonic of a StepRecovery diode driven with a doubled 100 MHz output of our hydrogen maser. A similar technique was used in Ref. [9]. The 2 MHz output from the mixer was filtered and directly counted with an Agilent 53132A counter with a 10 s gate time. This method was necessary to obtain a time series of the evolution of the beat between the cryoCSO and the microwave reference signal from our maser. See Fig. 8 for the fractional frequency offset from the moment the cryoCSO oscillator started up again. The drift rate is negative and decreasing. The best fit to 7 days of data is described by,
(5) 
where and days. By the 7th day Eq. (5) represents a decreasing fractional frequency drift in the cryoCSO frequency of /day. Since we observed the decreasing exponential frequency drift for many months from the initial startup, after cooling from room temperature, it cannot be attributed to a change in pressure in the helium gas space. That quickly stabilizes in a matter of hours. The cause of this long term exponential decrease is not known as yet.
The frequency stability calculated from the time series data is shown in Fig. 6 (curve (4)) with error bars. The exponential frequency drift (Eq. (5)) was removed before the Allan deviation was calculated. The multiplication process via the steprecovery diode used to generate the 11.2 GHz from the 100 MHz maser signal and amplification adds some noise, and also there seems to be an unknown temperature dependence there as well. Nevertheless, the stability calculated from the 7 days of data of Fig. 8 shows very good reproducability of the cryoCSO performance even after a 20 minute power interruption.
4 VLBI coherence
In VeryLongBaselineInterferometry radio astronomy, the coherence , a function of integration time , is defined as
(6) 
where is the phase difference between the two stations forming the interferometer. Considering only the phase difference due to the stability of the frequency standards used, the value of the coherence function for an integration time (henceforth referred to as coherence time) is a good figure of merit and can be estimated from, [17, 18]
(7) 
where is the angular frequency of the local oscillator (equal to the frequency of the astronomical source) and is the local reference signal Allan variance at integration time . The sum inside the square brackets should converge as , which would be true for frequency standards limited by white phase noise. With the standards used here this is not actually the case and the expression must be cut off at some point. In our case it was sufficient to use n = 3 at 345 GHz and n = 4 at 230 GHz and 100 GHz for calculations with all frequency standards. See Ref. [18] for further details.
For a particular standard at a given observing frequency, the coherence function has the range 0 to 1. is proportional to the fringe amplitude [21] and when = 1 there is no loss of coherence. The coherence time is approximately the time for which the coherence function approaches 1.
cryoCSO  cryoCSO  cryoCSO  cryoCSO  

(or T)  100 MHz  100 MHz  10 MHz  10 MHz 
0.99999  0.99991  
0.99994  0.99964  
0.99825  0.99513  
0.91664  0.84457  
0.39452  0.32815  
Maser  Maser  
(or T)  10/100 MHz  10/100 MHz  
0.98862  
0.95524  
0.82920  
0.47022  
0.16146 
From a qualitative point of view the coherence time is shorter at higher observing frequencies. Therefore the stability of the local reference will impact more strongly at higher frequencies over the shorter averaging times. Hence for millimeter wave observations the stability of the local oscillator becomes very important assuming the seeing conditions (the atmosphere) is not the limitation.
Using the analytical expressions, Eqs (3) and (4), obtained from fits to our measured stability data, in Fig. 7, we numerically calculated the coherence function from Eq. (7). Hence was determined using our 10 MHz and 100 MHz references, multiplied up to millimeter wave frequencies of 100 GHz, 230 GHz and 345 GHz. It is assumed that the same performance local oscillator is used on each end of the interferometer and that the multiplication process does not add any noise. The results are listed for 345 GHz in Table 2 along with the oscillator stability at fixed integration times and compared to that calculated from a very high performance maser. A typical value of the T4 Science maser [19] has been assumed. The frequency stability for that maser is shown as curve (4) in Fig. 7.
Similarly the coherence function was calculated using the stability data for this hydrogen maser (with the same stability at both 10 MHz and 100 MHz) where the same assumption was made as above. The resulting coherence function is then compared by plotting the ratio of derived from either the 10 MHz or 100 MHz synthesizer with that when the maser is used. The results are shown in Fig. 9, and show that the calculated coherence function is greater when using the 100 MHz reference due to its better short term stability.
The references synthesized from the cryoCSO are significantly more frequency stable than those from a hydrogen maser, especially the 100 MHz output. The latter offers improvements above 200% in the value of the coherence function at observing frequencies of 345 GHz at coherence times near s, assuming that the frequency reference is the limitation. It is apparent from Fig. 9 that at observing frequencies less than 100 GHz there is only a small advantage in using the cryoCSO. It must be said that where the decoherence effects of the atmosphere can be avoided, or compensated for, there is a clear advantage in integrating the signals for an hour or more. This is where the cryoCSO offers a big advantage over the hydrogen maser, especially for millimeter wave VLBI.
5 Conclusion
Two nominally identical frequency synthesizers based on low phase noise digital dividers and a direct digital synthesizer have been constructed and their performance evaluated. The reference signals at 10 MHz and 100 MHz were synthesized from a cryocooled sapphire oscillator and their phase noise and stability measured. The synthesizer residual single sideband phase noise, at 1 Hz offset, on 10 MHz and 100 MHz signals, respectively, were measured to be 135 dBc/Hz and 130 dBc/Hz. Their intrinsic frequency stability contributions, on the 10 MHz and 100 MHz signals, respectively, were measured as and , at 1 s integration time. As such the fractional frequency noise on 100 MHz output, at short integration times, is only about a factor of 2 greater than that of the cryocooled oscillator itself. The estimated total frequency stabilities of the new references are significantly better than those for the same output frequencies from a very high performance hydrogen maser.
From these measurements, we calculated the coherence function, a figure of merit, for millimeter wave VLBI radio astronomy. The references synthesized from the cryocooled sapphire oscillator offer improvements in terms of the coherence function of the order of 200% or more, where one is able to average the signal for several hours, at observing frequencies well above 100 GHz. The 100 MHz output produces a better result than the 10 MHz output, as might be expected. The cryocooled sapphire oscillator has the potential to replace the hydrogen maser as the low noise frequency stable reference for millimeter wave VLBI radio astronomy.
6 Acknowledgments
This work was made possible through an Australian Research Council grant LP0883292, with support from the University of Western Australia, CSIRO, Curtin University of Technology and Poseidon Scientific Instruments. The authors would like to thank A.E.E. Rogers and S. Doeleman for useful discussions and help, M.E. Tobar for useful suggestions, and especially M. Lours for providing some necessary control circuits.
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See www.t4science.com/documents/iMaser_Stability_3Cornered_Hat_
Method.jpg  See www.oscilloquartz.com/file/pdf/8607.pdf
 The measure of fringe amplitude and phase can be directly related to the (complex) Fourier component of the object brightness distribution that is under observation. While interference fringes contain both amplitude and phase information, most interferometric results published to date focus solely on the amplitude data. This is because atmospheric turbulence corrupts the observed fringe phases, rendering them almost useless by themselves. See www.vlti.org/events/assets/1/proceedings/1.5_Monnier.pdf for more information.