## Abstract

Digitally enhanced homodyne interferometry enables robust interferometric sensitivity to be achieved in an optically simple configuration by shifting optical complexity into the digital signal processing regime. We use digitally enhanced homodyne interferometry in a simple, all-fiber Michelson interferometer to achieve a frequency reference stability of better than 20 Hz/√Hz from 10 mHz to 1 Hz, satisfying, for the first time in an all fiber system, the stability requirements for the Gravity Recovery and Climate Experiment Follow On mission. In addition, we have demonstrated stability that satisfies the future mission objectives at frequencies down to 1 mHz. This frequency domain stability translates into a fractional Allan deviation of 3.3 × 10^{−17} for an integration time of 55 seconds.

© 2014 Optical Society of America

## 1. Introduction

Laser frequency stability is critical for many applications of precision metrology. Conventional interferometric methods use high finesse optical cavities with active feedback to stabilize laser frequency noise to the order of 10^{−15} depending on integration times [1–3]. For space-based missions such as the Gravity Recovery and Climate Experiment Follow On (GRACE Follow-On) mission, the cost of payload mass, as well as physical hardware space, are factors in the design of the metrology system. In recent years, fiber based interferometers have become competitive [4–6], especially when improved with post-processing using time delay interferometry (TDI) [7], and further enhanced with digital techniques [8].

Fiber based systems offer many advantages over optical cavities for implementation as an optical frequency reference, including compactness, mechanical robustness, ease of alignment and mode matching, reduced cleanliness requirements, whilst also being lightweight due to the fact that costly and bulky vacuum chambers are not needed. Furthermore, our fiber based system does not require the interferometer to be locked to a specific, resonant operating point as is required with optical cavities, allowing it to be tuned over a continuous frequency range.

Digital interferometry (DI) [9] has been shown to enhance various laser frequency-reference systems [8,10] and interferometric metrology systems [11,12]. DI offers benefits such as rejection of scattered noise, multiplexing of signals, and a simplification of optical hardware. The common feature of all DI techniques is the application of pseudo-random noise (PRN) codes to the interrogating signal. PRN codes are binary sequences of 0’s and 1’s, which can be used to apply a phase shift on the signal beam. The autocorrelation properties of these codes [13] allow the signals of interest to be isolated whilst rejecting spurious signals.

Digitally enhanced homodyne interferometry (DEHoI) [14] relies on a single spread spectrum signal beam consisting of two orthogonal PRN codes. These codes can be combined to produce a phase constellation as shown in Fig. 1, providing 4-level phase modulation. The purpose of these levels can be understood by analogy to conventional heterodyne phase measurements. In heterodyne interferometry, the phase of the beat note can be measured by recording the in-phase (I) and quadrature (Q) components. This is typically achieved by demodulating the detected signal with sinusoids that are 90 degrees out of phase. In homodyne DI, the I and Q components are obtained by shifting the phase of the light directly, using 4-level phase modulation separated by a quarter of a wave (90 degrees) corresponding to I, Q, -I, and -Q as shown in Fig. 1. The positive and negative values are obtained to retain compatibility with the range gating properties of conventional DI, as well as reduce sensitivity to modulation depth errors in the same way as for phase shifting interferometry [15].

The DEHoI technique uses the interference between two targets, one as a reference and the other as a genuine target, to generate a doubly modulated signal for photo-detection. By correctly matching the time delay for two cascaded decoders, the required signal, which is the optical phase difference between the target and reference, can be uniquely extracted. The time delay corresponds to the time of flight of the interrogation beam to the detector, and includes all associated electronic delays.

The homodyne system is the simplest optical configuration for digitally enhanced interferometry implementation as it does not require a dedicated, frequency-shifted local oscillator. Instead, the modulated reflection from the first target serves as the reference. This optical simplicity is achieved at the expense of increased signal processing complexity and demodulation algorithms - a convenient and cost effective trade-off. DEHoI retains full optical interferometric sensitivity while allowing the selection of targets based on time of flight, simultaneous detection of multiple targets and rejection of unwanted scattered light (such as fiber induced Rayleigh back-scatter). The homodyne configuration also allows the reference surface to be placed conveniently close to the target, thus eliminating common mode lead noise. In addition, fiber based interferometry is ideally suited to DI in general due to the relaxed chip length requirement and modest modulation/demodulation frequencies needed.

In this paper, we use DEHoI to interrogate an all-fiber Michelson interferometer. The interferometer is made sensitive to the laser frequency by introducing a 10 km arm length mismatch between the two interferometer arms. We demonstrate that DEHoI provides a high sensitivity, robust readout of the laser frequency, independent of the interferometer fringe condition and the absolute frequency of the laser being measured. In addition, we measure the optical path length stability of our all-fiber Michelson interferometer and demonstrate that this interferometer satisfies the stability requirements of the GRACE Follow-On mission. Although this target has been reached for optical cavity systems [16], this is the first time that an all fiber system has been shown to meet the GRACE Follow-On requirements across the measurement bandwidth of interest (the authors’ own results in [8] reached this target only between 20 mHz and 1 Hz). While other optical fiber frequency reference systems have demonstrated excellent stability, they have focused on short time scales and acoustic frequency ranges [4–6].

## 2. Experimental design

Figure 2 presents a schematic of our experiment. We phase modulated a 1550 nm Orbits Light-wave Ethernal single mode laser (with a linewidth of less than 200 Hz over 1 ms [17]) using an acousto-optic modulator (AOM). This modulated light source was split by a 50/50 fiber coupler into two optical fiber Michelson interferometers. Each interferometer was split into two pathways, or “arms”. One pathway directly returns the beam via a Faraday mirror with some attenuation, whilst the other pathway returns from another Faraday mirror after travelling through 10 km of fiber. The attenuator in the short arm was tuned to match the loss of the 10 km arm, ensuring optimal fringe visibility at the interferometer output by balancing the electric field vectors returned from the two arms.

In this experiment we have maximized the differential fiber interferometer arm length; limited by the round trip transit time and depolarisation of the return beam from the Faraday mirror of the long arm of the Michelson interferometer. This maximizes the response to laser frequency noise, minimizes detector dark noise and relaxes the required interferometric displacement sensitivity. With a 10 km arm length mismatch, and a frequency stability target of 30 Hz/√Hz, a relative stability requirement of 30 Hz/193 THz = 1.5 × 10^{−13}/√Hz, the displacement sensitivity required is relaxed to 1.5 × 10^{−13} × 20 × 10^{3} = 3 nm/√Hz. Therefore the required optical phase sensitivity is a modest 2*π* × 3 nm/1*μ*m = 18 milliradians/√Hz. This modest target, in turn enables the use of relatively short PRN codes (2^{11} − 1) with relaxed code orthogonality requirements.

The AOM modulated the laser light with a 4–level quadrature phase shift keying (QPSK) sequence. In this experiment, we used (2^{11} − 1) length codes. Our QPSK sequence uses a 4–level phase shifting sequence of ±*π*/4 or ±3*π*/4, constructed from two PRN codes (denoted as C_{1} for code 1 and C_{2} for code 2). Both codes are maximal length binary codes selected for their autocorrelation properties [13]. These codes are used to generate the in-phase and quadrature components as shown in the constellation diagram of Fig. 1. The logic used to generate the QPSK phase constellation of Fig. 1 is given by:

_{1}&C

_{2}’ denotes an ‘AND’ gate operation on code 1 and code 2, and ‘ $\overline{{\text{C}}_{1}}$’ denotes a ‘NOT’ gate operation on code 1.

Here *ϕ*_{QPSK} in Eq. (1) is used to modulate a numerically controlled oscillator (NCO) within the field-programmable gate array (FPGA) system, such that:

*ϕ*

_{RF}in Eq. (2), where

*f*

_{NCO}is the nominal frequency of the numerically controlled oscillator set to 40MHz) is fed into a high-speed digital-to-analogue converter (DAC), which is amplified before driving the AOM.

Although commercial QPSK electro-optic phase modulators are readily available, and have been successfully demonstrated in homodyne digital interferometry [14], here we use an AOM to provide an optical phase modulation as follows:

where*ϕ*

_{L}is the phase of the free running laser. The optical phase of the AOM (

*ϕ*

_{AOM}in Eq. (3)) therefore closely replicates the RF phase generated within our FPGA and accurately maintains the QPSK phase constellation of Fig. 1.

Our homodyne implementation of digital interferometry does not require an optical local oscillator field. Instead we rely on the interference between the two electric fields reflected from the arms of the Michelson interferometer. As each reflection is modulated by a QPSK sequence of two PRN codes, the interference signal between the two arms contains a doubly modulated QPSK sequence with four terms. In order to extract signals from the photodetector output voltage (V_{PD}) we need to perform two cascaded decoding operations. In Fig. 2 we explicitly show each decoder block; the first is fed with C_{1}(*τ*_{1}) and C_{2}(*τ*_{1}) whilst the second decoder is fed with C_{1}(*τ*_{2}) and C_{2}(*τ*_{2}). Each decoder block implements the following logic [14]:

*τ*is the chosen code delay in seconds.

Each decoder block outputs both I_{out} and Q_{out} such that two decoders, operating in parallel, are needed for the second operation, producing four signals: II, IQ, QI and QQ, which are the interference of the codes with delayed versions of those codes, and the interference between codes 1 and 2, or more explicitly:

These are gathered together to give the desired in-phase (I_{sig}) and quadrature (Q_{sig}) phase components of the signal of interest:

Finally, I_{sig} and Q_{sig} are combined to give

*ϕ*

_{sig}is the optical phase of the Michelson interferometer differential length. Figure 3(a) plots I

_{sig}and Q

_{sig}for a fiber interferometer with phase drifting due to laser frequency noise while Fig. 3(b) plots I

_{sig}and Q

_{sig}for an interferometer with a linear phase ramp. The phase is recovered for both these data sets using Eq. (12) and, after phase unwrapping, plotted in Fig. 3(c). The green trace shows interferometer phase drift while the red trace shows a linear phase ramp with a small amount of phase drift superimposed.

The linear phase ramp plotted in Fig. 3(c), red trace, is produced by linearly tuning the AOM frequency f_{NCO} of Eq. (2), and demonstrates a linear response to optical frequency independent of the interferometer fringe conditions shown in Fig. 3(b).

In order to measure the mechanical stability of the interferometers, we require a second identical interferometer to act as a reference. The differential measurement of the two interferometer outputs removes common mode noise sources such as laser frequency fluctuations. This differential measurement then provides us with a measure of the path length stability resulting from the fiber interferometers alone. However, due to length mismatches between the two interferometers, a simple subtraction of interferometer output phase signals suppresses the laser frequency noise by a factor of approximately one thousand only. In order to improve this, we utilize time delay interferometry (TDI) in post processing to compensate for the physical path length mismatch (approximately 5 m mismatch in a 10 km nominal arm length); see [8] for details of our TDI implementation. TDI processed data removes all traces of laser frequency noise and enables the underlying stability of the fiber interferometers to be measured.

## 3. Results

The cyclic error performance of our digitally enhanced homodyne interferometer system is presented in Fig. 3. The resulting phase, for both the drifting interferometer and the linear phase ramp, is plotted in Fig. 3(c) after unwrapping the raw optical phase. Finally, a Fourier frequency plot of the unwrapped optical phase data is shown in Fig. 3(d); the green trace is that of the drifting interferometer while the red trace is that of the linear phase ramp. The two traces are effectively identical except for the cyclic error components on the red trace at 5 Hz (fundamental) and 10 Hz (second harmonic). These cyclic error components correspond to free space displacement cyclic errors of 4 nm (0-peak) at 5 Hz and 3 nm (0-peak) at 10 Hz.

The free running laser frequency noise measured by both interferometers is plotted in Fig. 4 (pink and blue traces, “a”). The green trace, “b”, in Fig. 4 plots the phase difference between the two interferometers after TDI scaling has occurred. As can be seen, the combination of TDI scaling and differencing the phase outputs of the two interferometers results in a reduction in frequency noise of approximately five orders of magnitude at 5 mHz. Fig. 4 also plots the GRACE Follow-On mission requirements [18], “c”, shown as a solid red line (including future mission goals as a dashed red line) of 30 Hz/√Hz across the bandwidth of 10 mHz to 1 Hz. As shown in Fig. 4, the digitally enhanced homodyne readout in combination with TDI scaling meets this requirement across the entire bandwidth of interest for the GRACE Follow-On mission. This homodyne result is a factor of two to three better than our previous heterodyne result reported in [8].

The fractional Allan deviation of the interferometer frequency stability can be readily obtained by processing the differential interferometer time domain data. The interferometer fractional Allan deviation is plotted in Fig. 5 and shows that this deviation reaches a minimum of 3.3 × 10^{−17} at an integration time (*τ*) of 55 seconds. Shorter integration times show a 1/(*τ*) reduction corresponding to spectrally white noise above 20 mHz, as seen in Fig. 4 trace “b”. At integration times longer than 55 seconds, the fractional Allan deviation increases in proportion to *τ*. This long-term performance corresponds to the low frequency spectral region of trace “b” in Fig. 4, which below 20 mHz exhibits a 1/*f*^{2} dependence. Each fiber interferometer is housed in a two layered thermal shield designed to have a response time of 10 hours and a 1/*f*^{2} frequency response from a corner frequency of 1/(10 × 3600) = 27 *μ*Hz. Step function response testing of these enclosures indicates that the time constant is 10.5 hours and the frequency response does indeed roll off at 1/*f*^{2} as expected. The roll-off as 1/*f*^{2} in the low frequency region of the spectral density plot of Fig. 4(b) (green trace) indicates that the enclosure is performing as expected, and that the long term stability of each fiber interferometer is currently limited by the thermal environmental fluctuations after low-pass filtering of the enclosure. We therefore expect that any improvement in either the enclosure filter parameters, or active temperature control of the outside surface of the enclosures, should improve the long term interferometer stability accordingly.

## 4. Conclusion

We have shown that using digitally enhanced homodyne interferometry to interrogate an all-fiber Michelson interferometer can offer significant sensitivity improvements whilst minimising the optical complexity of the system. Our all-fiber interferometer system meets the frequency stability requirements for the GRACE Follow-On mission of 30 Hz/√Hz across the entire bandwidth of interest (10 mHz to 1 Hz). This system provides a highly sensitive, robust laser frequency reference that is dynamically tuneable and independent of both interferometer fringe conditions and the absolute frequency of the laser. Furthermore, our system uses passively stabilized thermal enclosures at atmospheric pressure in order to meet the GRACE Follow-On requirements, which offers a competitive advantage over systems requiring costly and bulky vacuum chambers.

Lastly, by providing laser frequency noise readout without the need for active control, our system can be used in feed forward signal processing techniques, such as TDI, to eliminate laser frequency noise without the constraints of actuator bandwidth limitations that apply to traditional feedback systems.

## Acknowledgments

This work was supported under the ARC DP grant number # DP140103575 and the Australian Governments Australian Space Research Programme.

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