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Abstract
We describe a ‘clock control unit’ based on a dual-axis cubic cavity (DACC) for the frequency stabilisation of lasers involved in a strontium optical lattice clock. The DACC, which ultimately targets deployment in space applications, provides a short-term stable reference for all auxiliary lasers—i.e. cooling, clear-out, and optical lattice—in a single multi-band cavity. Long-term cavity drift is compensated by a feed-forward scheme exploiting a fixed physical relation to an orthogonal second cavity axis; either by reference to an ultrastable 698 nm clock laser, or by exploiting the differential drift between orthogonal axes extracted by a single laser in common view. Via a change of mirror set in the cavity axis accessed by the clock laser, the system could also provide stabilisation for sub-Hz linewidths at the 698 nm clock wavelength, fulfilling all stabilisation requirements of the clock.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical cavity-stabilised lasers and optical clocks are unique tools for future tests of fundamental physics, advanced time and frequency metrology, and universal timescales e.g. for global navigation satellite systems (GNSS), navigation, and geodesy. The European Space Agency (ESA) is interested to invest in the next generation of clocks using high-finesse optical cavities and narrow optical transitions observed in cold atoms and ions. In particular, cubic cavity-stabilised lasers developed by the National Physical Laboratory (NPL) are possible references for the 2030s NASA/ESA LISA mission for gravitational wave detection from space [1]. They are also under development for the next generation gravity mission (NGGM) for laser interferometric monitoring of the Earth’s gravity field topology via orbiting satellite pairs [2]. The NPL-patented cubic cavities are also being developed for special relativity tests in space [3], and there is increasing interest for them in the “Kepler” satellite navigation augmentation for GNSS [4].
For future space optical clock (SOC) activity, there is a requirement to develop a robust system for laser frequency stabilisation and control at a number of specific wavelengths required to prepare atoms for clock interrogation. In particular, a stabilisation scheme for a strontium optical lattice clock (OLC) must tackle two stringent requirements: At a fractional target inaccuracy of $1\times 10^{-18}$, the absolute frequency of the confining 813 nm ’magic wavelength’ optical lattice must be controlled at the 100 kHz level [5]. Second, to reach ultracold atomic temperatures the second stage cooling laser, which operates on a 7.5 kHz intercombination line at 689 nm, requires both a short and long term frequency stabilisation at the 1 kHz level. The remaining lasers for cooling at 461 nm, often generated by second harmonic generation of a 922 nm seed laser, and further clear-out at 679 nm and 707 nm have comparably relaxed stability requirements at the 1 MHz level [6].
High accuracy optical clocks comprise a cold atom or ion-based physics package to provide the long-term frequency reference, a narrow linewidth optical clock laser, and a frequency comb to generate a microwave output [7]. The cooling, clear-out, and confinement lasers can typically have few kHz or greater linewidths although the clock laser needs a sub-Hz linewidth to take advantage of the narrow linewidth reference transition in the laser-cooled atoms or ion. Narrow laser linewidths are produced via a high-bandwidth servo to a high-finesse optical cavity [8]. The linewidth and short-term clock laser frequency stability requirements are linked to the clock target frequency stability and some compromises can be expected when compared against the best terrestrial clocks [9–12] that demonstrate low parts in $10^{16}$ single clock instability at 1 s. For a compact cavity, specifically the 5 cm NPL cube, the thermal noise limit will be $\sim 1\times 10^{-15}$ at 1 s for dielectric mirrors [13] and $<5\times 10^{-16}$ for crystalline mirror coatings [14]. Frequency control of the clock laser at longer times is provided by reference to the optical clock transition, with the frequency instability averaging the short-term laser noise as the square root of averaging time. In extended operation, this long-term control needs to be transferred at an appropriate level across a range of wavelengths to the auxiliary laser systems in the clock apparatus. Whilst a frequency comb could provide this frequency stability transfer, a multi-wavelength cavity system is a more straightforward solution that requires little in the way of additional hardware.
Optical cavities designed for multiple wavelength laser frequency stabilisation have been published previously, e.g. in [6,15–17], including cavities designed for space applications [18–22]. Within previous SOC activities [23], NPL was involved in the development of a robust Frequency Stabilization System (FSS) for controlling multiple laser frequencies [24]. For application in space, it is important to have a cavity design that is immune to environmental vibrations and we demonstrate here a ‘clock control unit’ (CCU) based on a dual axis implementation of the vibrationally insensitive NPL cubic cavity [25]. In contrast to a similar DACC for Ca$^+$ [26], our cavity could be used with either a strontium OLC or a Sr$^+$ ion clock [27]. For Sr$^+$, our DACC can be used for stabilisation at the 844 nm cooling sub-harmonic, 922 nm photo-ionisation sub-harmonic, and the 674 nm clock wavelength. Additionally, we propose new approaches for compensation of the cavity drift which take advantage of the fixed physical relation between orthogonal cavity axes. We show that anisotropy of the cavity spacer material makes it possible, following initial characterisation, to reduce drift without comparison to an external reference, thus providing a route to long-term stable cavity-based optical references suited to high-precision synchronisation and holdover. This holdover application could extend to situations where cavity drift correction is required during periods when the clock is not operational, e.g., when reloading an ion or servicing parts of the clock.
2. Multi-wavelength laser stabilisation
The NPL-patented cubic cavity was first presented in detail in [25], and introduced the concept of a force-insensitive tetrahedral mounting of a cubic cavity. The initial demonstration used a 5 cm ultra low expansion (ULE) glass spacer with a pair of optically contacted 12.7 mm diameter silica mirrors along a single axis. Here, we extend that concept to incorporate a second force-insensitive cavity axis on an orthogonal bore of a similar 5 cm ULE spacer. We use Corning 7973 ULE selected to have a zero linear coefficient of thermal expansion (CTE) at a convenient temperature above room temperature—typically between 25 $^{\circ }$C and 35 $^{\circ }$C. For mirror substrates manufactured from ULE, the assembled cavity will have a similar thermal tuning characteristic to the supplied ULE. Instead, we opt for fused silica mirror substrates which offer an approximate two fold reduction in the cavity thermal noise limit [13], with the caveat that the zero CTE temperature of the composite cavity is lowered substantially. A partial compensation is achieved by the addition of optically contacted ULE annuli to the reverse side of the mirrors [28]. Our assembled DACC is shown in Fig. 1, and comprises two pairs of 25.4 mm diameter, 1 m concave radius of curvature silica mirrors, each having this additional ULE thermally-compensating annulus. The mass added by the two pairs of 25.4 mm mirrors with compensating rings makes the cavity design less symmetric than in [25] but numerical modelling confirms it is still possible to find a null in the sensitivity to mounting forces through the truncation of the cube vertices to a specific depth. Adding a third set of mirrors with annuli might also aid in reducing sensitivity by increasing symmetry.
Fig. 1. Optically contacted DACC held with tetrahedral mounting in an aluminium frame. The precise positioning of the cavity within the centre of the frame is critical for achieving low acceleration sensitivity of the cavity.
All four mirrors of the DACC have an identical high-reflectivity bespoke multi-layer dielectric coating (provided by FiveNine Optics, Boulder) which is specified with finesse $>10$ k at the critical 698 nm and 689 nm wavelengths, corresponding to a cavity fringe full width half maximum (FWHM) < 300 kHz. At 813 nm, where there is a more relaxed frequency stability requirement, a finesse (fringe FWHM) of $>2$ k (1.5 MHz) was specified. The cavity is mounted in an aluminium frame by four tetrahedrally arranged nylon spheres, enclosed by a single layer polished aluminium heat-shield, and housed in a resistively heated aluminium vacuum chamber. The chamber is maintained at close to 25 $^{\circ }$C with a few mK instability and control bandwidth $\sim$10 mHz, and a pressure $<1\times 10^{-7}$ mbar by a 2 l/s ion pump.
All lasers except the clock laser are multiplexed in single-mode PM63-U40D fibre to propagate along a single ‘AUX’ cavity axis with the clock laser fed to the orthogonal ‘CLK’ axis, see Fig. 2. Both axes accommodate a free-space optical isolator (OI), which is broadband in the AUX axis (BBOI), to prevent parasitic etalons forming between the high-reflectivity cavity mirrors and upstream optical elements. Losses through the multiplexed arm up to the cavity input are in the range of 18 to 27 dB, depending on wavelength, and 12 dB for the CLK axis. Since the laser power requirements at the cavity are typically only a few tens of micro-watts, the losses are not problematic. Lasers are frequency referenced to the cavity fringes using the Pound-Drever-Hall (PDH) locking method in an electronic sideband modulation scheme [24,29]. This provides for each wavelength channel a tuneable offset $\nu _{\mathrm {off}}^{\mathrm {ch}}$ between the desired laser carrier frequency and cavity fringe by means of an optical sideband which is phase modulated to generate the PDH error signal. Analog Devices AD9910 direct digital synthesis (DDS) chips are employed to generate the phase modulated rf drive signals for each wavelength which are applied to the light through individual fibre pig-tailed waveguide electro-optic phase modulators (EOMs). Where possible the EOMs were chosen to be of the annealed proton exchange waveguide type [30] as they support the propagation of only one linear polarisation [31] leading to levels of residual amplitude modulation (RAM) which are low and stable. Active RAM suppression is possible using titanium in-diffused waveguide modulators [32].
Fig. 2. Schematic depiction of the multi-channel multi-wavelength stabilisation module designed for use in Sr or Sr$^+$ clocks. The system exploits the physical relation between orthogonal dual-axis cubic cavity (DACC) axes (AUX and CLK) to remove cavity drift. BS is a 50:50 beamsplitter; MM are fibre collimators that provide cavity mode matching; OI is an optical isolator; BBOI is a broadband optical isolator; WDM is a wavelength division multiplexer made up of four approximately 50:50 fibre splitters labelled A, B, C, and D. We implement an electronic-sideband modulation scheme [29] using a waveguide electro-optic modulator (EOM) driven by a AD9910 direct digital synthesis (DDS) chip for each optical wavelength channel. A digital microcontroller-based servo with proportional, integral, and differential (PID) gain, tracks the drift of the CLK axis and updates each DDS via a frequency tuning word (FTW). Each channel is phase modulated (PM) for Pound-Drever-Hall (PDH) locking by cycling through a phase tuning lookup table stored in memory (RAM) on-board each DDS, with the trigger of this forming the local oscillator (LO) for demodulation of the PDH error signals, which are detected by a single rf photodetector on each axis. The DDS are clocked at 1 GHz by a phase-locked loop (PLL) referenced to an external (EXT) 10 MHz source. The laser lock status is monitored by dc photodetectors (labelled ‘trans’) centred on the cavity transmission. Independent frequency stabilisation servos are implemented for each AUX channel. The complete optical assembly of the CCU including the DACC, pictured bottom right, is contained within a $456\times 306\times 166$ cm package; volume 23.2 l, total mass 15.9 kg. The electronics are housed in a separate 3U 19" unit.
An on-board state machine in the AD9910 allows for the rapid cycling of pre-programmed phase offset words to generate a phase modulated waveform, the period of which is made available externally and is subsequently processed through a programmable phase delay chip to provide the demodulation local oscillator (LO) for the PDH lock. PDH error signals are detected by a single photodiode in the AUX axis and separated during demodulation by use of discrete modulation frequencies; 8.900 MHz, 6.940 MHz, 16.625 MHz, and 12.500 MHz for the locks at 689 nm, 698 nm, 813 nm, and 922 nm, respectively, chosen to minimise cross-talk while allowing sufficient clear bandwidth for the locking servo. The DDS chips are clocked by a 1 GHz phase-locked loop (PLL) with 10 MHz reference clock input. To bridge the 1.5 GHz half free-spectral-range (FSR) of the 5 cm DACC, each DDS is frequency doubled and amplified to provide $\nu _{\mathrm {off}}^{\mathrm {ch}}$ up to 800 MHz. If a larger offset is required, up to the full half FSR, then the modulation depth is optimised for the second order sideband to which the cavity lock is made.
An Arduino microcontroller sets parameters of the DDS and phase shifting chips and is configured via a python script. The microcontroller also runs a low-bandwidth digital PDH lock which is implemented on the CLK axis with feedback to $\nu _{\mathrm {off}}^{\mathrm {clk}}$ via the DDS frequency tuning word. This provides a high resolution frequency record of the offset between the reference light which is coupled to the CLK axis, and the cavity resonance.
To compensate drift in the AUX axis, we make use of the highly correlated creep of the orthogonal CLK and AUX axes arising due to their fixed physical relation. The measured CLK axis correction is independently scaled according to wavelength as $r_{\lambda }=\lambda _{\mathrm {clk}}/\lambda _{\mathrm {aux}}$ and fed-forward to $\nu _{\mathrm {off}}^{\mathrm {ch}}$ of the corresponding auxiliary channel. An overall scaling factor, $a$, is included to account for the difference in rate of the correlated CLK and AUX isothermal drift arising from anisotropy of the ULE glass spacer, such that the total scaling is $c=ar_{\lambda }$. This scheme removes the need for an additional probe of the AUX axis for compensation of drift under certain conditions.
3. Characterisation and drift control
3.1 System properties
The multi-wavelength aspect of the CCU was assessed using a sub-set of lasers at 689 nm, 813 nm, and 922 nm. PDH error signals were measured by simultaneous scanning of lasers across cavity resonances, shown in Fig. 2, and were observed to have no significant cross-talk. The PDH discriminator determines the sensitivity of the error signal to cavity length fluctuations and is determined in part by the cavity finesse, listed in Table 1. Short-term stability was assessed by a beat note comparison at selected wavelengths against independently stabilised lasers. We observed combined linewidths of a few kHz at 689 nm—which has the most stringent linewidth requirement—and a 1 s instability of $\sim$200 Hz in the lock of the cavity offset to an independently stabilised 698 nm sub-hertz clock laser.
Table 1. CCU system properties. IL; insertion loss from input fibre to cavity input mirror, FWHM; full width half maximum of the Lorentzian cavity fringe-width. Properties of auxiliary channels 2 and 3, intended for use at the repump wavelengths 679 nm and 707 nm, were not measured and so are not included.
The coating resilience was also tested by irradiation on a 36 MeV proton beam line at the University of Birmingham to a total fluence of $4\times 10^{10}$ protons/cm$^2$; targeted along the cavity axis, centred on the mirror through a single vacuum viewport. No notable degradation of the mirror performance was observed in later measurements.
The dc acceleration sensitivity of the DACC was measured simultaneously for orthogonal cavities through a series of inversions of the CCU module in x, y, and z directions, totalling 2 g acceleration change per axis, while observing the required frequency correction to maintain lock to an ultrastable reference laser at 698 nm. The results, given in Table 2, are higher than in [25] in the directions along each orthogonal cavity axis; since low acceleration sensitivity was not a focus of this project we have not investigated this further.
Table 2. DACC measured dc acceleration sensitivities in fractional frequency units per unit acceleration, g. The CLK and AUX cavity axes are defined along coordinate axes x and y, respectively.
The cubic cavity, mounted within its aluminium frame with single-layer heat-shield, demonstrates a thermal time constant of 8.5 hours in response to a small temperature step. The CTE zero temperature was determined for both the CLK and AUX axes by observing the deviation of resonant frequencies in response to a temperature sweep, and gave results of approximately 24 $^{\circ }$C and 26 $^{\circ }$C respectively and prevents the simultaneous operation of both cavity axes at the optimal CTE zero temperature.
3.2 Drift and compensation
To characterise and compensate frequency drift of the DACC under conditions approaching isothermal creep, we stabilised a single ultrastable 698 nm laser simultaneously to both CLK and AUX cavities; using the described Arduino-based digital lock to track the CLK axis, and an additional synthesiser-based analogue lock to the AUX axis. The absolute frequency of each DACC axis, $\nu ^{\mathrm {clk}}_{\mathrm {abs}}$ and $\nu ^{\mathrm {aux}}_{\mathrm {abs}}$, was determined by the sum of the counted cavity-offset frequency $\nu ^{\mathrm {clk}}_{\mathrm {off}}$ and $\nu ^{\mathrm {aux}}_{\mathrm {off}}$, respectively, together with the absolute frequency of the 698 nm reference laser measured by a hydrogen maser-referenced optical frequency comb. The evolution of these frequencies was tracked over a ten week period and is plotted in Fig. 3(a). We find these drifts to be strongly correlated ($r=0.9997$, see Fig. 3(b)), providing a stable value of the isothermal drift ratio [Eq. (2)] of $a=0.71$. This value is consistent with that of a previous data set taken three years prior when the cavity was first installed in vacuum, despite the absolute drift rate having reduced by a factor of around 25. Accounting for a total of three data sets, with another taken in 2020, we arrive at a value of $a=0.70(3)$ where we estimate the standard uncertainty by taking the semi-range around a time weighted mean$/\sqrt {3}$.
Fig. 3. (a) Measurement of the DACC drift by reference to a 698 nm ultrastable laser over a ten week period. Linear drift rates of the CLK and AUX axes were measured to be 29 mHz/s and 20 mHz/s, respectively, and are observed to be highly correlated, inset (b). Drift is compensated using both CLK axis- and differential drift- feed-forward schemes, as described in the text, together with a straightforward linear drift compensation. The residual frequency deviations after compensation are shown in (c). The overlapping Allan deviation is calculated for each of these schemes and is compared with the uncompensated DACC instability in (d).
The observed asymmetry most likely arises from anisotropy in the ULE glass with the material properties along the growth axis differing from those in the transverse directions. The ULE anisotropy is still being fully investigated but we expect that forming cavities on different pairs of faces of a ULE cube would lead to significantly different values of $a$. The DACC reported here was manufactured with one of the two cavities parallel to the growth axis such that the asymmetry is maximised. With both cavities orthogonal to the growth axis we expect this to reach a minimum (i.e. with $a\sim 1$). Assuming isothermal drift, the frequency difference between orthogonal cavity modes, $\nu _{\mathrm {diff}}$, can be described by the following equations:
We exploit these relations in two feed-forward schemes to compensate the drift of the DACC. The first approach relies on an external reference to measure the drift of the CLK axis; this is ultimately imagined to be the atomic reference of an optical clock in which a laser locked to the CLK axis serves as the interrogating probe laser. With the factor $a$ predetermined, Eq. (2) can then be used to compensate the AUX axis drift, as shown in Fig. 3 as AUX (feed-forward compensated). This removes the need for any additional hardware around the AUX axis for monitoring of its drift. The second scheme removes the need for an external reference altogether, following a determination of $a$, and instead relies on the ULE anisotropy, such that $a\neq 1$ and $\Delta \nu _{\mathrm {diff}}\neq 0$. The DACC drift is then extracted using Eq. (1) and Eq. (2) with a measure of $\Delta \nu _{\mathrm {diff}}$ which is straightforwardly obtained by a single hitherto unstablised laser in common view of both DACC axes. Using a feed-forward arrangement the frequency deviations of both CLK and AUX axes are therefore removed but the correction involves the measured differential drift inflated by factors $1/(1-a)$ and $a/(1-a)$ respectively. This suggests that there could be an advantage in orienting the cavity bores to maximise the difference in creep between both orthogonal axes, with a theoretical but not practical optimum at $a=-1$.
The overlapping Allan deviation frequency instabilities of the CLK and AUX axes with and without drift compensation are calculated and plotted in Fig. 3(d). A residual frequency instability of $<500$ Hz at $10^6$ s is achieved for the compensated AUX axis in the measured CLK axis feed-forward scheme. This instability is inflated by a factor 3.4 for the AUX axis where only the differential drift is used, plotted as “AUX (differential compensated)” in Fig. 3. We compare these compensation schemes to a simple linear drift removal for the AUX axis which shows a slightly degraded performance compared to the CLK axis feed-forward which additionally suppresses common mode noise from, e.g., temperature fluctuations. Since noise in tracking of the cavity drift exceeds the value of the drift at short timescales ($<10^3$ s), this unnecessarily increases the short-term frequency instability when compensating using the measured differential drift at 1 s intervals, as done here. Implementing low-pass filtering of the drift correction signal could help to retain the short-term frequency instability offered by the uncompensated cavity axis.
For a 5 cm room temperature cavity we should reach a flicker frequency noise floor of around 0.5 Hz ($10^{-15}$ fractional frequency), corresponding mainly to the thermal noise contributed by dielectric mirror coatings. For the thicker coating stack of our multi-band mirrors we might expect this to increase slightly, however, the 150 Hz noise floor demonstrated here results from excess noise in the laser stabilisation, which manages a little better than $10^{-3}$ precision on a cavity fringe of roughly 300 kHz full-width-half-maximum. Increasing the cavity finesse 30 fold to 300 k with a dedicated mirror set at 698 nm should provide a corresponding reduction in laser linewidth, with the remainder required to reach the thermal noise floor overcome by routine improvements to the locking setup.
4. Future development
Development of the DACC as a space compliant ultrastable reference cavity is being pursued in a parallel set of work. A critical requirement is to increase the technology readiness level (TRL) of the cubic cavity and mounting arrangement which must survive the rigours of launch while subsequently achieving low acceleration sensitivity. However further measurements and design changes will clearly also enhance the laboratory device.
Overall system performance could be improved by additional in-vacuum heat-shielding and improved characterisation of the CTE. We plan to explore the option of operating at a set temperature where the ratio of linear CTE of the two bores equals the isothermal drift ratio $a$. Although this temperature is not expected to be where the CTE is zero for either bore, it will mean that frequency deviations arising from both thermal and isothermal fluctuations can be removed indiscriminately in application of the discussed drift compensation schemes involving $a$. Beyond the effects of ULE asymmetry, it should also be possible to tune CTE zero temperatures by adjusting dimensions of the ULE compensating annuli, adding further to the possibilities here.
Critically, if employing the differential drift compensation scheme, $a$ must prove to be a better predictor of cavity drift over extended measurement periods and across multiple batches of ULE than an extended model of a single cavity axis creep, which is typically well approximated by a constant linear drift plus exponential relaxation term [33]. However, reliance on such a model provides no probe of thermally driven frequency deviations, which have been tackled elsewhere in measure and feed-forward systems using temperature sensors external to the cavity [34].
Finally, replacing the CLK cavity axis mirrors with high-finesse alternatives, the same cavity could provide a reference for the ultrastable clock laser with thermal noise limited instability at the level of $1\times 10^{-15}$ for dielectric mirror coatings and $<5\times 10^{-16}$ if crystalline mirrors are employed [14].
5. Conclusion
We have described and characterised a compact ‘clock control unit’ based on a rigidly held dual axis cubic cavity configured to serve the laser stabilisation requirements of a strontium OLC. The device makes use of a low-loss multi-band dielectric mirror coating to allow all necessary wavelengths to be referenced along a single cavity axis, and could be adapted to accommodate other systems based on e.g. neutral Yb, Ca, Hg, Cd, Mg, or ions Yb$^{+}$ and Ca$^{+}$.
By referencing each orthogonal cavity to an ultrastable laser at 698 nm we have shown that the observed isothermal drift is highly correlated and therefore suited to a feed-forward compensation scheme involving measurement in only one of the cavity axes. We also found the drift to be unequal, with a ratio of $0.70(3)$ observed over a period of 3 years, revealing of an anisotropy of the ULE which exists between the ULE growth axis, along which we have one of our cavities, and transverse directions. The anisotropy is also likely responsible for the observed 2 K difference in the temperatures where the CTE is measured to be zero for each cavity axis. If instead we were to form both orthogonal cavities along transverse directions, the drift ratio would be expected to approach unity.
The ULE anisotropy was exploited to compensate drift in both CLK and AUX cavity axes by combining a calibration of the fixed drift ratio with a running measurement of only the differential drift between axes, which was achieved using a single laser in common view. If the dual axis cavity was specialised to operate with mirrors supporting a high-finesse on each axis for a single wavelength, we could expect the drift compensation to improve correspondingly. With better control of temperature through implementing multiple stages of in-vacuum radiation shielding, or through tuning the ratio of CTEs for each axis to match the isothermal ratio, we could see further improvement, putting the long term stability of such a system in the realm of $<10^{-14}$ fractional frequency instability, comparable to that achieved by independent reference cells. Since the drift ratio used in this scheme essentially takes the role of the external frequency reference and is determined by properties of the ULE material, which may vary over time, further investigation is required to establish its long-term utility as a stand-alone frequency reference.
Funding
European Space Agency (4000117290/16/NL/BJ, 4000132316/20/NL/AR/z).
Acknowledgment
We are grateful to the University of Birmingham for providing access to their proton beam line for radiation compliance testing, and to Uwe Sterr at PTB for carrying out functional testing of the system. We thank Marco Schioppo for reading of the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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