Ultraprecision intersatellite laser interferometry

Precise measurement of relative change in the distance between two spacecraft Δd using a laser metrology system is typically achieved by measuring the optical phase change Δϕ of the laser beam propagating along the line of sight of two spacecraft, which is expressed by

$$\begin{equation}\Delta d = \lambda \frac{{\Delta \varphi \,}}{{2\pi }} = \frac{c}{v}\frac{{\Delta \varphi \,}}{{2\pi }},\end{equation} \tag{ 1 }$$

where λ and ν are the wavelength and frequency of the laser, and c is the velocity of light. According to equation (1), laser frequency fluctuation is one of the sources of noise measurement. The frequency instability of a laser can be characterized by the linewidth, the Allan variance in the time domain, and the power spectral density (or amplitude spectral density) in the frequency domain [7, 8]. The laser linewidth is a measurement of the frequency resolution in spectroscopy. A narrow linewidth represents good monochromaticity and a highly temporal coherence in laser interferometers. The Allan variance, describing the mean value of the momentary relative frequency fluctuation averaged over a time interval, characterizes the temporal stability of laser frequency. The power spectral density (PSD) of laser frequency noise is the Fourier transform of the square of the laser frequency noise. A more commonly used description is its square root. The amplitude spectral density (ASD) characterizes the amplitude distribution of frequency noise with the unit Hz/Hz^1/2.

For laser interferometry, the spectral density is more advantageous and informative for indicating the laser frequency stability because the scientific target signals are expected to be within a specific frequency band of interest. The total noise measurement can be deduced from the laser frequency noise spectral density by integrating the spectral density within the frequency band of interest.

For an interferometer with unequal arm length L, the displacement measurement error caused by the laser frequency fluctuation can be expressed as,

$$\begin{equation}\frac{{\Delta d}}{L} = \frac{{\Delta v}}{v}.\end{equation} \tag{ 2 }$$

For space-based laser interferometry, a laser with a 1064 nm wavelength is preferred because of the low absorption of the space medium at this wavelength through the intersatellite propagation. High power output is essential for adequate received signal light power at the remote spacecraft in order to reduce the shot noise of interference measurement. In addition to these two issues, the maturity of a frequency-stabilized laser system must also be taken into account.

The interferometer of the LISA has arm lengths of 2.5 ∼ 3 × 10⁹ m [9]. The required frequency noise of the laser is 30 Hz/Hz^1/2 from 10 mHz to 1 Hz at a prestabilized state [10] by means of the Pound–Drever–Hall (PDH) method. The laser frequency noise is then further suppressed successively by using the arm-locking technique [11] and time delay interferometry (TDI) [12] to a level of μHz/Hz^1/2, which arises from the requirement for measurement noise of 10 pm/Hz^1/2 with 1% relative arm length variation. The residual intensity noise (RIN) of the laser in the LISA should be better than 3 × 10⁻⁸/Hz^1/2 at the beat frequency and twice the beat frequency and <2 × 10⁻⁴/Hz^1/2 from 0.1 mHz to 1 Hz as well [10]. The latter requirement is for reducing light pressure fluctuation applied on the test masses during measuring of the position of the test mass by optical readout.

The TianQin is also a space-based gravitational-wave detection mission, having a nearly equilateral triangle constellation of three geocentric orbiting spacecraft that are about 1.7 × 10⁵ km away from each other. The laser frequency noise is required to be less than 10 Hz/Hz^1/2 at 10 mHz after laser frequency prestabilization and then is further suppressed by TDI to a level of 0.1 mHz/Hz^1/2 to meet the requirement for the measurement noise of 0.5 pm/Hz^1/2 with 1% arm length mismatch (≈2000 km).

Space-qualified lasers must meet the following requirements: (1) the size, weight, and power consumption must be constrained; in principle, the smaller the better; (2) the laser system must be qualified by mechanical, thermal, electromagnetic interference (EMI), and high-energy radiation compatibility tests; (3) all operations of the laser must be automatic onboard; and (4) the laser must be reliable without failure or obvious degradation during the designed lifetime of the space missions.

An Nd:YAG solid-state laser with nonplanar ring oscillator (NPRO) structure [13] and 1064 nm wavelength has been tested for reliability in terms of intrinsic frequency and intensity stability, mechanical robustness, power efficiency, and large tunable frequency range. Space-qualified Nd:YAG NPRO lasers have been developed during the last decades and are commercially available. These lasers have passed all of the qualification tests and have operated successfully in space for the laser communication terminals of TerraSAR [14, 15], the LISA Technology Package of LISA Pathfinder (LPF) [16], and the first intersatellite laser interferometer of Gravity Recovery And Climate Experiment–Follow On mission (GRACE-FO) [17].

Up to a few watts of laser power is required for the application of gravitational-wave detection in space, e.g. LISA and TianQin (see table 1). Promising solutions for generating high laser power of up to 4 W has been realized, including a stand-alone NPRO laser with special mechanical and pumping designs and a master oscillator fiber amplifier (MOFA) configuration that is based on the high-power fiber amplifier seeded by a low-power stabilized master laser oscillator [18], shown in figure 2.

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A fiber-based laser is also suitable for space applications due to its robustness and compactness. The suitability of a commercial Yb-doped distributed feedback (DFB) fiber laser and its amplifier system emitting 1 W of power for the LISA mission has been proposed [19, 20]. Following that, a fiber ring laser that has a comparable free-running frequency noise and a smaller RIN, compared to a Nd:YAG NPRO laser at a low frequency band, was developed for the LISA mission [21]. This fiber ring laser can achieve a faster frequency tuning (up to 10 MHz of bandwidth) than commercial DFB or distributed Bragg reflector (DBR) lasers by using an intracavity phase modulator [22]. Although these Yb fiber lasers perform comparably to Nd:YAG NPRO lasers at a low frequency band, they all show a large RIN peak around 1 MHz due to their typical relaxation oscillation, which could affect the heterodyne interferometry near this frequency.

Recently, the planar-waveguide external cavity diode laser (PW-ECL) has become another promising candidate for a space-qualified master laser because of advantages that include lower cost, less consumption, and more robust design of the butterfly package, as well as better stability compared to other lasers. The PW-ECL laser, which consists of a semiconductor gain chip and a Bragg reflector laser cavity on the planar waveguide, was commercially available for the telecom C-band (around a wavelength of 1550 nm). Its suitability for precision measurements in space has been studied [23, 24]. A 1064 nm version of the commercial PW-ECL laser was successfully built, and a full MOFA system with 2.5 W of output power has shown good stability and reliability results for the LISA mission [25].

A long lifetime is also a critical requirement for a space-qualified laser system, which depends on the reliability of pumping diodes for both the seed laser and fiber amplifier. A highly reliable laser diode pumping unit should enable a steady output of power, good hermetical performance, and redundancy. In addition, compact size, robust mechanical structure, and flexible electrical and optical interfaces are also needed for a space-qualified pumping diode. Over the last two decades, space-qualified pumping modules based on multiple bars or arrays of laser diodes with fiber-coupled, high-power output have been developed [26]; and a 10 W diode pumping module with a reliability of 0.9998 over a ten-year lifetime has been reported [27].

A laser frequency stabilization system consists of a frequency-tunable laser, an ultrastable frequency reference, a feedback control electronic system, and compulsory optical and electrical links between the laser and frequency reference.

The PDH technique, invented in 1983 [28], has become the most popular method for laser frequency stabilization by means of locking it to the resonance frequency of an ultrastable FP cavity. The resonances of an FP cavity occur when the roundtrip distance is an integer multiple of laser wavelength, and the linewidth of resonance is represented by the cavity finesse that is determined by the reflectivity of cavity mirrors.

Figure 3(a) shows a basic setup for the PDH laser frequency stabilization, in which the electro-optic phase modulation (EOM) is used to produce a pair of Bessel sidebands with equal amplitude and opposite phase on both sides of the laser frequency before they are incident into the FP cavity. When the laser frequency is near the cavity resonance frequency, the signals reflected from the cavity's front mirror form an asymmetric dispersion curve after demodulation by a mixer, as shown in figure 3(b). This is used as the error signal for feedback control of the laser frequency. We will skip the detailed working principle and theoretical analysis about the PDH method. For readers who are interested, please refer to [28, 29].

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An ultrastable reference cavity for space applications usually features a high-symmetric geometry and a rigid mounting design for the purpose of acceleration and orientation insensitivity. Moreover, the rigidly mounted optics to maintain a steady mode-matching efficiency between the laser and a certain transverse electromagnetic (TEM) mode of the FP cavity is necessary, as illustrated in figure 3(a).

The space-qualified PDH laser stabilization system for the GRACE-FO mission was developed by the Jet Propulsion Laboratory (JPL) [30] with a high-finesse, tapered structure commercial cavity, a mode-matching optical bench, and a fiber-based optical layout for phase modulation. A frequency stability of 30 Hz/Hz^1/2 at 0.01 Hz was achieved (shown as a solid black line in figure 4). This system was launched and is working successfully for the GRACE-FO mission [17].

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The high stability laser (HSL) system for the next generation gravity mission (NGGM) has been established by the National Physical Laboratory (NPL) [31] and has achieved a laser frequency stability of better than 20 Hz/Hz^1/2 at a frequency above 0.01 Hz, shown as the solid blue line in figure 4. This system is based on the PDH scheme with an ultrastable cubic cavity mounted symmetrically in a tetrahedral configuration inside a compact vacuum chamber [38].

To meet the needs of the TianQin mission and the next generation gravity mission in China, our group has been studying and developing a space-based PDH laser frequency stabilization system since 2013. We have constructed a prototype of a PDH laser frequency stabilization system [39] in which the FP cavity and its mode-matching optics are firmly integrated on an ultrastable monolithic optical bench using the hydroxide-catalysis bonding technique [40]. The digital control program realized on the field-programmable gate array (FPGA) platform was developed based on the dual-loop scanning and real-time analyzing algorithms [41]. Its functions include initial laser frequency calibration and automatic locking and relocking. The preliminary test shows that the laser frequency stability is better than 30 Hz/Hz^1/2 at a frequency band ranging from 0.7 Hz to 10 Hz.

In addition to the PDH method, the hyperfine transition of molecular iodine is another well-known absolute reference for laser frequency stabilization [42]. In this method, a 1064 nm laser is frequency doubled and locked to an absorbing hyperfine transition of iodine by the Doppler-free spectroscopy technique. In principle, laser frequency locking to the molecular iodine transition can offer an exact absolute frequency reference without thermal drift. It may simplify the initial heterodyne beat note acquisition process and thus has attracted more and more attention to study its potential applications for space-based laser interferometry [43, 44].

The best fractional frequency stability of the iodine-transition-based laser stabilization setup reaching the flicker floor at about 4 × 10⁻¹⁵ was realized in the laboratory [45] by a four-pass optical scheme and active control of residual amplitude modulation. The first flight-like absolute laser frequency reference for future space missions was developed based on a quasi-monolithic ¹²⁷I₂ optical setup by the modulation transfer spectroscopy technique [32]. In this setup, the optical layouts are jointed into one integrated glass baseplate using the space-qualified adhesive bonding technique [46]; and the frequency stability achieved meets the requirements for LISA and NGGM, as shown by the solid green line in figure 4. The improved setup based on this design was successfully flown on a sounding rocket in 2018 [47].

According to the requirements of space-based gravitational-wave detection missions, the noise level of phase measurement is 1 ∼ 10 μrad/Hz^1/2, corresponding to 0.1 ∼ 1 pm/Hz^1/2 for a laser wavelength of 1064 nm. For the GRACE-FO and NGGM missions, a measurement noise of about 1 mrad/Hz^1/2 will be enough to meet the requirements.

The output of the photodetector in a heterodyne laser interferometer is a beat-note signal, and the beat-note frequency is determined by the Doppler effect caused by the relative motion between two spacecraft. The information on gravitational waves or the Earth's gravity field is contained in the phase of this beat note. A phasemeter is used to measure the phase difference between the beat note and the reference signal provided by an onboard ultrastable oscillator (USO).

For spaceborne gravitational-wave detection missions, the typical requirements for a phasemeter are: (1) the resolution of phasemeter must achieve 10 μrad/√Hz from 0.01 Hz to 1 Hz. (2) The phasemeter must have a large measurement range due to the Doppler frequency shift of ∼10 MHz caused by intersatellite relative motion. (3) The functions of automatic (laser) frequency acquisition and adjustable gain are needed because the amplitude and frequency of the beat note are unknown initially. Moreover, the phasemeter must provide a high data sampling rate for weak-light phase locking, which is discussed in the next section.

There are many types of phase measurement methods. The zero-crossing method was studied in [48–50]. The frequency of the measured signal can be as large as 20 MHz, and the resolution is about 0.01°. The major noise comes from the fluctuation of the reference voltage of the zero crossing.

The method based on phase-locked loop (PLL) is a closed-loop scheme for measuring phase. It has a high resolution and a large dynamic range. Since the frequency shift (caused by the Doppler effect) and the phase change (caused by the change in optical path length) of the beat note can be recorded, the displacement and the (intersatellite) relative speed can be obtained simultaneously. Therefore, the PLL-type phasemeter is the optimum method for intersatellite laser interferometry.

The research groups in the Joint Institute for Laboratory Astrophysics (JILA) [48–50], JPL [51–53], Germany Albert Einstein Institution (AEI) [54–59], Lulea University of Technology (LUT) [60], TianQin (Sun Yat-sen University and Huazhong University of Science and Technology) [61–63], and the National Microgravity Laboratory, Chinese Academy of Sciences (NML CAS) [64, 65], have gradually developed phasemeters since 2000. The method of zero crossing (JILA 2001, 2003, 2006, weak-light interferometer signal mixed to about 10 kHz) was replaced by the digital PLL method. For a signal frequency range of 1 mHz to 1 Hz, the noise level of the phasemeter has already achieved less than 1 μrad/Hz^1/2, which can meet the requirement of spaceborne gravitational-wave detection missions. Schwarze et al developed a picometer-level, stable, hexagonal optical bench to verify the LISA phasemeter [59]. However, the noise level of phase measurement at a low frequency band (0.1 ∼ 10 mHz) requires further study, especially in the case of weak light (100 pW) received and large heterodyne frequency shift (20 Hz s⁻¹). Data on the development and performance of precision phasemeters published thus far are summarized in figure 5.

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The schematic diagram of a typical phasemeter based on the digital phase-locked loop (DPLL) algorithm is shown in figure 6. The analog input signal (i.e. the beat note of the heterodyne laser interferometer) is digitized by the sampling circuit that consists of a signal conditioning circuit (i.e. an antialias filter or an amplifier) and an analog-to-digital converter (ADC). This digitized signal is then multiplied by the sinusoidal signal generated by the numeric-controlled oscillator (NCO) to obtain the phase error. The signal of the phase error is then sent to a proportional-integral (PI) feedback controller, of which the control signal is used to keep phase synchronization between the NCO and the input beat note. To lock the phase of a signal with an unknown frequency, a frequency detector (counter) is used to perform automatic acquisition within a certain tracking range. Once the NCO is phase locked to the input signal, the frequency and the phase change of the NCO can be recorded, which correspond to the speed and the displacement of the measured target, respectively. All of the above processes can be realized either by using digital hardware (e.g. FPGA) or analog electronics.

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Although a collimated laser beam is used in intersatellite laser interferometry, the beam size of the emitting laser propagating through a long intersatellite distance becomes much larger than that of the satellite. This is due to the divergence angle of laser beams determined by the diffraction limit and a very long propagation distance of about 10⁵ ∼ 10⁶ kilometers [66]. As a result, the light power received by the target satellite is very weak, and the power attenuation of the intersatellite laser beam is so serious that the typical scheme of a Michelson interferometer with only reflecting mirrors is not suitable to intersatellite laser interferometry. To solve this problem, an optical phase-locking technique is used, instead of a simple reflection. The optical phase-locked loop is able to lock the phase of the local (slave) laser to that of the light received. In this way, the light emitting from the slave laser has an identical phase to the received light but a much stronger light power.

The Doppler effect, caused by the relative motion between two satellites, is another issue needing to be considered. Although the satellites' orbits are analyzed and optimized to build up a stable triangle constellation, the relative motion between two satellites still varies periodically with a maximum speed of about 10 m s⁻¹, which will result in a Doppler frequency shift of 10 MHz [67, 68]. Therefore, a heterodyne phase-locked loop with an offset frequency of about 15 ∼ 20 MHz must be used. Moreover, the long-term stability and the low frequency residual noises of the weak-light phase locking are critical, especially for detecting low-frequency gravitational-wave signals.

The challenge of weak-light phase locking is to achieve ultrahigh precision phase measurements with interference signals of very small signal-to-noise ratio (SNR). Assume that the emitting laser power is 1 W, the intersatellite distance is 1 million kilometers, the diameter of telescope D is 30 cm, and the beam radius on the emitting telescope is ${{{\omega }}_0} = 0.45D$. According to the far-field diffraction of a Gaussian laser beam ${\theta _0} = {{\lambda }}/{{\pi }}{{{\omega }}_0}$ and the beam quality ${{\text{M}}^2} = 1.2$, the divergence angle can be expressed as $\theta = {{\text{M}}^2}{{\lambda }}/{{\pi }}{{{\omega }}_0}$ [69]. The beam diameter at the receiving spacecraft is $2L\theta \approx 6\;{\text{km}}$; therefore, a telescope of 30 cm receives only 2.5 ppb of the laser power emitted. In addition to power attenuation due to long-distance propagation, the power loss of all optical components along the optical path of the interferometer must be taken into account, so that the effective light powers detected by photodetectors are even smaller [70].

Regarding the early development of weak-light phase locking, Enloe and Rodda [71] used a homodyne phase-locking technique, in 1965, to make two single frequency helium–neon (He–Ne) lasers to oscillate at exactly the same frequency with a residual phase difference of less than a third of one degree. After the 1990s, the optical phase-locked loop has been widely used in the areas of optical communication, quantum optics, cold atom physics, and laser interferometers [72–76]. This research established the foundation for the optical phase-locking technique. However, the research on weak-light phase locking faces new challenges, including quantum shot noise and phase noise at low frequency, ranging from 0.1 mHz to 1 Hz.

Liao et al demonstrated homodyne phase locking to 2 pW incoming weak light. The laser light was attenuated by using neutral density filters, and the balanced detection method was used to eliminate the laser intensity noise and improve the signal-to-noise ratio. The experimental results showed that the phase error was 290 mrad (rms) over a locking duration of 1.5 min [77].

Ye and Hall demonstrated a high-performance optical phase-locked loop between two continuous wave Nd:YAG lasers. The feedback system employed the laser's internal piezoelectric transducer (PZT) and an external acousto-optic modulator (AOM). The PZT (with the bandwidth of about 20 kHz) corrected slow but potentially large laser frequency drifts, and the AOM (with the bandwidth of about 200 kHz) eliminated fast frequency fluctuations. A residual phase noise of about 1 μrad was achieved, and the out-of-loop phase noise was approximately 0.6 μrad/Hz^1/2 at 1 Hz and above [78].

McNamara et al reported an experiment to evaluate a weak-light phase-locking scheme in which two independent, diode-pumped Nd:YAG NPRO lasers were locked together with a 15 MHz frequency offset. Using a weak light intensity of 17 nW, they obtained the smallest residual noise over the frequency range of 10 ∼ 800 Hz [79]. Based on this work, they improved the performance of weak-light phase locking for LISA in 2005. The experimental results showed that the residual phase noise of the slave laser reached the shot noise limit (0.13 mrad/Hz^1/2) above 0.4 Hz when the power of the weak light was attenuated down to 13 pW [66].

Diekmann et al implemented an analog optical phase-locked loop with an offset frequency of 20 MHz, in which the detected light powers of two lasers were 31 pW and 200 μW. The phase noise between two lasers was twice the shot noise limit down to 60 mHz, and the residual phase noise below 60 mHz mainly came from the analog electronics of the phasemeter. This paper gives some details on the noises generated by the photodetectors, such as the shot noise of the photodiode, the Johnson noise, and the equivalent current noise of the transimpedance amplifier [68].

Dick et al [80] and Francis et al [81] demonstrated weak-light phase locking with light powers down to 40 fW and 30 fW, respectively. In order to lock two lasers with such weak light powers, the parameters of the loop filter have to be optimized to balance laser phase noise and shot noise, such that the cycle slip rate can be minimized.

The noises of weak-light phase locking come from three major components of a typical phase-locked loop: laser, phasemeter, and photodetector. The sources of laser noise include frequency fluctuation and photon number fluctuation (shot noise). Since the shot noise is a quantum noise limit, an optimized design of the phase-locked loop must ensure that the shot noise is dominant. In the presence of only shot noise, cycle slips in a phase-locked loop can be reduced to any desired level by reducing the loop bandwidth [82]. However, the loop bandwidth should be optimized when the laser frequency (phase) noise cannot be ignored.

The measurement noise of the phasemeter is caused by the sampling time jitter of the analog-to-digital convertor (ADC), the quantization error of the ADC, and the thermal drifts of some thermal-sensitive electronic components. The phase error caused by the sampling time jitter of the ADC can be reduced by using the pilot-tone correction technique [62, 63].

Principally speaking, reducing the in-loop noise requires a high control bandwidth. In contrast, reducing the out-loop noise requires a low control bandwidth. Therefore, the control bandwidth (or the loop gain) is a key parameter that should be optimized for weak-light phase locking. In addition, the loop (time) delay is another key parameter that should be carefully considered. According to the previous work [72], the control bandwidth should be determined by considering the loop delay and the laser frequency noise (i.e. the laser linewidth).