1. Introduction

Low noise optical fibre interferometry is a key requirement for strain sensing [1, 2], fibre optic gyroscopes used in avionics [3] and as a proposed frequency reference for low earth orbit geodesy missions [4]. The increased interest in optical fibre for these interferometric sensing applications is due to several advantages over comparable free space optical systems such as the ability to easily build mechanically robust optical configurations that are typically lightweight, compact, remain free of contaminants and easy to align and mode match.

Fibre optic interferometry however, is strongly polarisation dependent as it relies on the coherent mixing of two signals. This can lead to signal fading due to the random fluctuations in the state of polarisation of the interfering beams and also poor visibility [5]. This polarisation induced fading has been shown to also produce phase noise in the demodulated sensor output [6]. Although Faraday isolators offer a convenient solution [7] they are sensitive to variations in magnetic field strength and are limited to interferometers with specific topologies. Faraday isolators are also susceptible to vibration as they are assembled from free space optical components. These issues with Faraday isolators are exacerbated where there are very long lengths of fibre, when there are coupling elements that depend on the polarisation state of the input beam [8] and in environments of varying magnetic field strength or high mechanical stress, for example aerospace applications.

A proven way of simplifying hardware issues and associated maintenance requirements of optical fibre interferometric systems [9] is the use of digital interferometry [10]. Digital Interferometry (DI) uses pseudorandom noise (PRN) codes, binary codes with autocorrelation properties, to code a pseudorandom phase shift on the signal of interest. After detection signal coherence can be recovered with synchronous decoding while scatter and noise are rejected. Furthermore digitally enhanced homodyne interferometry (DEHoI) [11] uses two PRN codes to implement quadrature phase shift keying (QPSK) and generate the in-phase (I) and quadrature (Q) components to recover the beat note phase and eliminates the need for the separate local oscillator required in heterodyne experiments. See [9] for full QPSK signal processing details.

In this work we analyse the interferometer polarisation noise with DEHoI in a 10 km bidirectional fibre loop interferometer configuration free of Faraday mirrors. The 10 km fibre improves frequency discrimination and DEHoI enables the orthogonal polarisation components to be referenced with unique QPSK codes allowing us to track the evolution of the polarisation noise and algebraically cancel it using real time processing. This technique allows a reduction of polarisation noise to the level of approximately $10\text{Hz}/\sqrt{\text{Hz}}$, that is currently limited by the double Rayleigh backscatter (RBS), where the double RBS is due to secondary reflections from refractive index defects in the fiber.

2. Background

The Jones matrix formalism allows a complete description of the polarisation for coherent light and has been previously used to introduce the concept of the eigenstate of polarisation [12]. For a length of optical fibre the incident and output light can be represented with the transformation equation E_out = AE_in, where the matrix elements are defined in the following way

The standard matrix identities, AA⁻¹ = A⁻¹A = I, allow us to write E_in = A⁻¹ E_out where the inverse matrix (A⁻¹) describes the polarisation process travelling in the opposite direction along the fibre. As AA⁻¹ = I this reverse direction contradicts all the polarisation changes imposed by the fibre on the forward pass and therefore recreates the input polarisation state with a common phase and amplitude

The use of DEHoI creates a doubly modulated signal for photodetection that is a result of the interference between a reference path and the signal path. By selecting the appropriate digital delay line taps we match the required time delays for the two cascaded decoders. The optical phase difference between the signal and the reference (local oscillator) can then be uniquely extracted. The QPSK code allows us to label each of the orthogonal polarisations at the input of the fibre and track the evolution of the polarisation through the interferometer. Specifically, as shown in Fig. 1, to measure the phase of (det(A)) for each interferometer (IF2 in the direction of the orange arrows and IF1 in the reverse direction indicated by green arrows) we inject orthogonally coded fields $E_{\text{xi}}^{\mathrm{C}1}$ and $E_{\text{yi}}^{\mathrm{C}2}$. Here C1 and C2 are independent field programmable gate array (FPGA) generated 4-level QPSK logic codes assigned to each of the orthogonal polarisation states. The same orthogonal code is used for the local oscillator (LO) and the signal to give doubly modulated outputs with respective time delays τ_LO and τ_sig. Therefore the Jones matrix contains two cross-coupled components (b and c) and two independent elements (a and d)

 

Fig. 1 Simplified architecture of the experiment. PBC: polarising beam combiner, 3 dB: 50/50 fibre coupler, P: fibre polarisation adjustment paddles, T: low back reflection fibre termination,10 km; 10 km single mode fibre, AOM (40 MHz, 80 MHz): acousto-optic modulator with 40 MHz, 80 MHz resonant frequency, C1, C2: independent 4-level FPGA generated QPSK logic, τ_lo,τ_sig: digital delay lines coded into the FPGA to decode the LO and signal paths respectively, PD1, 2: photodetector1, 2, ϕ_IF1,2: recovered phase noise from interferometer 1, 2.

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3. Experimental design

The design of our experiment and the signal processing architecture is illustrated schematically in Fig. 1. The fibre interferometer has the topology of a loop delay line that is simultaneously operated in both the forward and reverse directions using a narrow linewidth laser (< 200 Hz over 1 ms) [13]. Here the reference path is straight through the 3dB input/output coupler on the 10 km loop, while the signal path includes a single transit around the 10km loop in addition to the straight through path. The doubly modulated/encoded interference is recorded at the photodetector and subsequently decoded to recover the optical phase. As the reference path is common to both the signal and the reference fields, the optical phase recovered records only the phase of the 10km loop. The physical time delay is set by 10 km of single mode fibre on the signal path coupled to the straight through (LO) path with a 3 dB coupler. This 10 km mismatch between the reference and signal path increases the interferometer sensitivity to laser frequency noise. The bidirectional operation produces a frequency reference free from any gyroscopic phase shifts when the two directions are added and also allows the removal of common mode laser frequency noise and most mechanical noise when the two directions are subtracted. The removal of this noise is necessary to allow investigation of the much smaller polarisation noise that nevertheless compromises interferometer performance. The orthogonal polarisation components E_xi and E_yi are set with manual “paddle” fibre polarisation controllers and “labelled” with a unique QPSK code before recombination at the polarising beam combiner (PBC) and subsequent injection into the fibre interferometer. The QPSK phase modulation is generated in a LabView FPGA system and is applied with two acousto-optic modulators (AOM’s) each with a resonant frequency of 40 MHz. The doubly modulated signals are recorded on both photodetectors and decoded with the time delays and codes appropriate for the LO polarisation and signal polarisation required. Therefore code orthogonality alone allows us to recover the polarisation information, eliminating the need for a polarisation analyser at the receiver.

A 2 Hz sawtooth phase ramp with a peak amplitude of 2π, to scan a full free spectral range, is implemented on both of the 40 MHz AOM’s to frequency shift the 1/(N) code noise, (where N is the code length of 2047 chips) beyond the signal bandwidth of interest, see [14] for details. There is an 80 MHz AOM in IF2 to provide frequency shifts in the path of one interferometer with respect to the reverse path of the other interferometer eliminating single pass RBS from the measurements. This AOM also allows us to introduce an arbitrary signal for independent calibration.

The interferometer was put in a hermitically sealed enclosure with foam insulation to damp acoustic vibrations. Additional thermal insulation with a 40dB/decade roll off and a time constant of 10 hours was added to reduce interferometer measurement noise caused by temperature fluctuations[15].

4. Results

Figure 2 plots Δa = arg(a1) − arg(a2), Δb = arg(b1) − arg(b2), Δc = arg(c1) − arg(c2), Δ(d) = arg(d1) − arg(d2) and Δϕ = ϕ1 − ϕ2. Here ϕ1 and ϕ2 are the optical phase measurements of the determinants of interferometers 1 and 2 respectively. We manually manipulate the polarisation of one interferometer for the 10 minute duration of the experiment and inject a 38 mHz signal with an amplitude of 200 Hz via the 80 MHz AOM in interferometer IF2. The phase signal shown in figure 2(i) verifies that large scale length changes are common mode to both interferometers. Figure 2(ii) shows that while individual matrix elements were strongly effected by polarisation noise, exhibiting large phase wander and discrete 180 degree phase flips, the determinant phase eliminates all polarisation noise and returns the signal phase only, with the associated 38 mHz modulation visible in the time domain of figure 2(iii).

 

Fig. 2 (i): Tracking the phase evolution of each interferometer, IF1 (red dashed line) and IF2 (magenta), as the polarisation drifts over the course of 10 minutes. (ii): Tracking the total phase evolution and the phase evolution of the matrix elements of the difference between the two interferometers of the above plot. Δa (red), Δb (green), Δc (violet) and Δd (blue). Note the independent polarisation matrix elements (Δa and Δd) have less phase noise than the cross coupled components (Δb and Δc). (iii): Enlarged section of (ii) around 100 s clearly showing recovery of the 38 mHz signal.

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Figure 3 shows the normalised amplitudes of both the determinants and the individual Jones matrix elements for each interferometer (IF1 figure 3(i); IF2 figure 3(ii)). The data for each interferometer is taken simultaneously and the polarisation for IF2 is manually adjusted over the full polarisation ellipse many times over the 10 minute data run. There is a clear anti correlation between the independent polarisation components (a and d) and cross coupled components (b and c). Furthermore the measurements show the determinant amplitude is maintained at a large and approximately constant amplitude while individual element amplitudes completely fade due to random polarisation noise injected into IF2.

 

Fig. 3 (i) IF1: Tracking the normalised amplitude changes as the polarisation for IF1 drifts over 10 minutes. a₁ (red), b₁ (green), c₁ (violet), d₁ (blue) and IF1 (black). (ii) IF2: The polarisation for IF2 is manually adjusted in real time as the data for the above plot is recorded. a₂ (red), b₂ (green), c₂ (violet), d₂ (blue) and IF2 (black).

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Figure 4 shows a Fourier spectral density plot of this time data. The polarisation noise suppression is barely visible in each interferometer trace when there is no common mode subtraction. However when the subtraction of both interferometers are used to suppress common mode noise sources polarisation noise below 1Hz is clearly visible. This polarisation noise is rejected when det(IF1)-det(IF2) is processed to reveal the signal at 38 mHz with a SNR of 40dB. This polarisation sensitivity det(IF1)-det(IF2) is limited by double RBS noise as described in [14].

 

Fig. 4 Traces ν_IF1 and ν_a1 respectively show the total frequency noise for the determinant and the matrix element “a” of (for example) the forward interferometer path and look the same on this scale over much of the frequency range. However for the subtraction between the two interferometer paths there is a clear difference in the polarisation noise of the matrix elements ν_a1−a2 and the determinant ν_IF1−IF2. Note that this plot is normalised to 1 Hz, with a measurement time of 600 s and a bandwidth of 1/600 Hz. Therefore the applied 200 Hz signal deviation at 38 mHz is scaled to give $5000\text{Hz}/\sqrt{\text{Hz}}$. Shaded areas represent polarisation suppression and the spectra have been averaged above 100 mHz (dashed lines).

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Figure 5 shows the amplitude spectral density of a 6 hour data run with no deliberate polarisation state manipulation. Small amounts of polarisation noise are visible in a1-a2 while det(IF1)-det(IF2) rejects this noise down to the double RBS background. Note that by changing the length of the fiber interferometer within the chamber, we were able to demonstrate that the polarisation noise of a1-a2 is approximately constant; indicating that this noise source is from the fiber components external to the chamber rather than from within the fiber interferometer itself. From the 6 hours of data with an observed noise floor of $10\text{Hz}/\sqrt{\text{Hz}}$ we can estimate the interferometer phase stability. With a bin width of 4.63 × 10⁻⁵ Hz and a FSR of 20 kHz the phase stability is calculated to be 20 × 10⁻⁶ radians/6 hours.

 

Fig. 5 (i): Averaged traces for the six hour data run. Traces ν_IF1 and ν_a1 respectively show the total frequency noise for the determinant and the matrix element “a” of (for example) the forward interferometer path. The 2 Hz phase ramp to frequency shift the code noise has been overlaid from the original (unfiltered) data. Subtraction of common mode noise between the two interferometer paths is shown for Jones matrix element a: Δν_a1−a2 and for the determinant of the two interferometer paths (ν_IF1−IF2). (ii): Original (unfiltered) data at low frequency showing a noise floor of $\approx 10\text{Hz}/\sqrt{\text{Hz}}$ and the 30 mHz FM signal with a peak deviation of 20 Hz. The yellow shaded region illustrates the polarisation noise.

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5. Conclusion

Here we use the determinant phase to measure the average optical path length allowing rejection of polarisation noise down to double RBS noise. This method has no tuning parameters to optimise, is simple, robust and well suited to remote and harsh field operation. However this technique also gives access to all the components of the Jones matrix, with interferometric precision in real time, potentially allowing full use of the Jones matrix for precision ellipsometry analysis of optical components. Furthermore, the use of DEHoI allows this ellipsiometry to be implemented without any polarisation optics after beam preparation. In addition to the elimination of polarisation noise the DEHoI technology used here is not affected by code noise and allows the removal of single pass RBS. These improvements should enable the use of much longer fibre lengths and therefore better frequency discrimination and potentially new metrology applications.