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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 20 — Sep. 24, 2012
  • pp: 22195–22207
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Digitally enhanced homodyne interferometry

Andrew J. Sutton, Oliver Gerberding, Gerhard Heinzel, and Daniel A. Shaddock  »View Author Affiliations


Optics Express, Vol. 20, Issue 20, pp. 22195-22207 (2012)
http://dx.doi.org/10.1364/OE.20.022195


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Abstract

We present two variations of a novel interferometry technique capable of simultaneously measuring multiple targets with high sensitivity. The technique performs a homodyne phase measurement by application of a four point phase shifting algorithm, with pseudo-random switching between points to allow multiplexed measurement based upon propagation delay alone. By multiplexing measurements and shifting complexity into signal processing, both variants realise significant complexity reductions over comparable methods. The first variant performs a typical coherent detection with a dedicated reference field and achieves a displacement noise floor 0.8pm/√Hz above 50Hz. The second allows for removal of the dedicated reference, resulting in further simplifications and improved low frequency performance with a 1pm/√Hz noise floor measured down to 20Hz. These results represent the most sensitive measurement performed using this style of interferometry whilst simultaneously reducing the electro-optic footprint.

© 2012 OSA

1. Introduction

Optical interferometry is widely used for sub-wavelength metrology in areas such as gravitational wave detection [1

1. The LIGO Scientific Collaboration, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Rep. Prog. Phys. 72, 076901 (2009).

], fibre sensing [2

2. T. T.-Y. Lam, M. B. Gray, D. A. Shaddock, D. E. McClelland, and J. H. Chow, “Subfrequency noise signal extraction in fiber-optic strain sensors using postprocessing,” Opt. Lett. 37, 2169–2171 (2012). [CrossRef]

] and geodesy [3

3. B. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. Klipstein, and W. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J Geodesy 10.1007/s00190-012-0566-3 (2012). [CrossRef]

]. When an application requires high sensitivity over a large working range, such as interferometer lock acquisition [4

4. M. Evans, N. Mavalvala, P. Fritschel, R. Bork, B. Bhawal, R. Gustafson, W. Kells, M. Landry, D. Sigg, R. Weiss, S. Whitcomb, and H. Yamamoto, “Lock acquisition of a gravitational-wave interferometer,” Opt. Lett. 27, 598– 600 (2002). [CrossRef]

] or co-phasing of multi-element telescopes [5

5. A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

], continuous phase tracking techniques such as heterodyne interferometry are commonly applied [6

6. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993). [CrossRef]

]. State-of-the-art heterodyne interferometry implementations have demonstrated picometer-scale displacement sensitivities over timescales of thousands of seconds [7

7. G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the Laser Interferometer Space Antenna,” Phys. Rev. Lett. 104, 1–4 (2010). [CrossRef]

, 8

8. S. Anza, M. Armano, E. Balaguer, M. Benedetti, C. Boatella, P. Bosetti, D. Bortoluzzi, N. Brandt, C. Braxmaier, M. Caldwell, L. Carbone, A. Cavalleri, A. Ciccolella, I. Cristofolini, M. Cruise, M. Da Lio, K. Danzmann, D. Desiderio, R. Dolesi, N. Dunbar, W. Fichter, C. Garcia, E. Garcia-Berro, A. F. Garcia Marin, R. Gerndt, A. Gianolio, D. Giardini, R. Gruenagel, A. Hammesfahr, G. Heinzel, J. Hough, D. Hoyl, M. Hueller, O. Jennrich, U. Johann, S. Kemble, C. Killow, D. Kolbe, M. Landgraf, A. Lobo, V. Lorizzo, D. Mance, K. Middleton, F. Nappo, M. Nofrarias, G. Racca, J. Ramos, D. Robertson, M. Sallusti, M. Sandford, J. Sanjuan, P. Sarra, A. Selig, D. Shaul, D. Smart, M. Smit, L. Stagnaro, T. Sumner, C. Tirabassi, S. Tobin, S. Vitale, V. Wand, H. Ward, W. J. Weber, and P. Zweifel, “The LTP experiment on the LISA Pathfinder mission,” Classical Quant. Grav. 22, S125 (2005). [CrossRef]

]. This extreme sensitivity comes at the cost of complexity and drives stringent stability and scattering requirements across the system, making widespread adoption of interferometric metrology difficult outside of the laboratory. This paper proposes a novel metrology technique, Digitally Enhanced Homodyne Interferometry, which augments homodyne interferometry with pseudo-random phase modulation to provide high sensitivity with large dynamic range and reduced system complexity.

In heterodyne interferometry, the signal beam is offset in frequency from the local oscillator field causing their relative phase to evolve over time. Phasemeters commonly extract this relative phase by determining the in-phase and quadrature components of the resulting beat note [18

18. D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc. 873, 654–660 (2006). [CrossRef]

, 19

19. I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009). [CrossRef]

]. DEHoI uses a phase modulator, rather than a frequency offset, to shift the relative phase of the LO and signal beams. Deriving inspiration from phase shifting interferometry [20

20. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996). [CrossRef]

22

22. E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011). [CrossRef]

], we choose 4 levels of phase, which produce the equivalent of in-phase and quadrature components. By using high-speed phase shifts driven by a pseudo-random noise code, we receive the range-gating benefits of DEHeI without the need for a second laser frequency or additional modulation [23

23. G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010). [CrossRef]

].

In this paper we provide an introduction and demonstration of DEHoI, including the specific pseudo-random phase stepping used for phase estimation. After introducing the phase estimation method, an analysis of the multiplexing capabilities are presented. Two fibre interferometer configurations are used to interrogate multiple round-trips of a fibre ring, providing a self-consistency mechanism for verifying the displacement sensitivity and multiplexing capabilities. Both implementations realise displacement noise floors of order 1pm/√Hz over low audio frequencies.

2. Digitally Enhanced Homodyne Interferometry

Digitally Enhanced Homodyne Interferometry (DEHoI) provides high sensitivity and complexity reduction by adapting the DEHeI techniques to a homodyne detection. We present the 4 point pseudo-random phase stepping algorithm to be used for phase estimation along with two possible realisations. The first technique employs an architecture similar to that of DEHeI, where a discrete LO and signal beams are combined prior to photodetection. The second variant of DEHoI takes advantage of the homodyne nature of the measurement to do away with a separate LO path, thereby realising a further reduction in system complexity.

2.1. Phase measurement

DEHoI infers the relative phase between a LO and signal beams by determining the in-phase (I) and quadrature (Q) components of the interference, the ratio of which allows the determination of relative phase. The measurement of these components is obtained by application of a 4 point {0,π /2,π, 3π /2} rad phase stepping pattern which maps out four equally spaced points along the interference fringe as shown by Fig. 1(a). The use of four steps provides unique measurement values for a given phase offset whilst also granting suppression of random measurement errors and systematic offsets [24

24. P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 5–7 (1987).

]. By utilising phase steps where cos(φs)= ±1 and sin(φs)= 0, or cos(φs)= 0 and sin(φs)= ±1, the I and Q components are directly sampled and do not need to be inferred from the measurement.

Fig. 1 (a) shows the interference fringe (dotted line) F = cos(φφS) for a phase offset φ and time-varying phase shift φS. A heterodyne offset frequency Δf produces a linear phase shift φS(t)= 2π Δft, mapping out the fringe over T = 1/Δf sec. Also shown are the cosine in-phase (I) and sine quadrature (Q) decompositions used for phase estimation. Discrete phase steps of φS(t) = {0,π /2,π, 3π /2} rad will result in the phase stepped (solid) response which directly samples the I and Q amplitudes. (b) Depicts the complex phasor “symbol” for each phase step resulting from the modulation sequences cI[n] and cQ[n].

This style of four stepped phase modulation may be directly generated by modulating the real and imaginary quadratures of an optical field, hence it is referred to as Quadrature Phase Shift Keying (QPSK) [25

25. J. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994).

]. With two input sequences cI[n],cQ[n] = ±1 for modulation onto a field IN, the n’th sample of the modulated field can be written as E˜m[n]=12(cI[n]+icQ[n])E˜IN, creating the four possible phase “points” as shown in Fig. 1(b). Note that this QPSK definition is rotated π /4rad from the nominal case without detriment to the measurement method. When this modulated field is interfered with the Local OscillatorLO, the measured power envelope on a photodetector P = is given by:
P(t,n)=E˜IN(t)2+E˜LO(t)2+E˜LO(t)E˜IN(t)2(cI[n]cos(φd)+cQ[n]sin(φd))
(1)

In the absence of noise or errors, the trivial phase estimate φ̃d from symbols s[n] = P(t,n), as defined in Fig. 1(b), is;
I^=s[1]+s[4]s[2]s[3]22E˜LoE˜INcos(φd)Q^=s[1]+s[2]s[3]s[4]22E˜LoE˜INsin(φd)}φ^d=arctan(Q^I^)
(2)

Dedicated modulators, such a Dual-Parallel Mach-Zehnder (DPMZ), are capable of generating optical QPSK modulation once the modulator is locked at the correct operating point [26

26. P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006). [CrossRef]

]. In contrast to telecommunications systems, which may use QPSK modulation to transfer data, DEHoI uses knowledge of the applied modulation to apply coherent phase estimation methods such as the Phase Locked Loop [27

27. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol. 27, 901–914 (2009). [CrossRef]

].

2.2. Multiplexing by PRN modulation

DEHoI uses two Pseudo-Random Noise (PRN) sequences for QPSK modulation, randomising the phase stepping pattern and introducing a propagation delay dependance to the measurement. The PRN sequences used are typically Maximum Length Sequences (M Sequences), which are periodic, ±1 valued sequences with low auto- and cross-correlation properties [10

10. S. Golomb, “On the classification of balanced binary sequences of period 2n-1(Corresp.),” IEEE Trans. Inform. Theory 26, 730–732 (1980). [CrossRef]

]. These correlation properties allow isolation of individual paths based upon propagation delay. DEHoI targets a specific signal (with unique propagation delay) by correlating the received signal against an appropriately delayed copy of the modulation sequence. All signals arriving with different delays will be poorly correlated, allowing isolation of the desired signal. When averaged over the length of the PRN sequence N, the contribution of undesired paths will be suppressed by 𝒪(1/N), providing significant suppression of spurious signals. To successfully isolate a signal, it must be separated by at least the period of one phase step. With modulation speeds in excess of 1GHz available, the resulting component separations of < 10cm will allow application of this technique in compact/bench-top systems.

Figure 2 shows the fibre ring interferometer implemented in Section 2.3. This setup was selected as a simple test case for demonstration of the delay multiplexing and measurement sensitivity of the technique, with each round trip of the ring treated as an independent sensor path for evaluation. In this setup, the cI and cQ PRN sequences are generated and modulated onto the metrology field through a DPMZ modulator. The field enters a fibre ring formed from a 50/50 coupler and an Electro-Optic Modulator (EOM), which provides a calibration signal for testing. The i’th pass through the ring will partially transmit (αi ≪ 1) a component of the metrology field that is encoded with the cumulative ring phase φi φRING × iRound Trips. The transmitted fields are then interfered with a Local Oscillator (LO) for detection at PDS. Each round trip signal will arrive with a relative propagation of delay τi = 2Li/cn sec, for ring length Lim and effective speed of light cn. The implemented parameters are listed in Table 2.3, however it is sufficient to note for now that the relative propagation delay for each ring round trip is greater than the period of one modulation symbol which permits separation.

Fig. 2 Showing a proposed fibre fibre ring interferometer setup. The laser is split into LO and metrology branches. The Metrology branch is modulated with I and Q PRN sequences using a Dual Parallel Mach-Zender modulator. A small ( 5%) pick-off to detector PDC is used for modulator control. The system under test is a fibre ring, formed using a 50/50 coupler with a fibre connected EOM for calibration signal injection. The ring output is interfered with the LO at detector PDS for detection. The ring will create a set of distinct ‘round-trip’ signals Si, with decaying amplitude due to losses, which will be separable based upon different propagation delay τi. This is illustrated by the ring impulse response, which plots signal strength against propagation delay.

In addition to the cumulative ring phase φi, each component of the metrology field will acquire contributions from the lead φL and return fibres φR, whilst the LO will have phase φLO. With appropriate delays included, the phase of the i’th ring round trip will be;
φ^i(t)=φL(tτiτR)+φi(tτR)+φR(t)φLo(tτLO)
(3)

Ignoring the constant power terms, the power envelope of the interfered fields Int detected at PDS is given below. For now, we assume all signal-signal interactions are significantly lower in power than the signal-LO components as ‖LO‖ ≫ ‖αim.
P˜Int(t)i=0Nαi2E˜LO(t)E˜IN(tτi)[cI(tτi)cosφ˜i(t)+cQ(tτi)sinφ˜i(t)]

The PRN correlation properties can now be used to extract each round trip of the fibre ring. To isolate the real component j(t) of the j’th round trip phase we perform a correlation (★) by multiplying the detected signal with cI(tτj) and averaging over the sequence repetition period Tc. By matching the demodulation and propagation delays, the desired cos φ̃j(t) term is recovered as cI(tτj)cI(tτj)= 1 for all t, whilst the other terms are suppressed by the cross- and auto-correlation coupling factors Rτi,τj=1/TctTCtcI(tτj)cQ(tτi)dt, and Aτi,τj=1/TctTCtcI(tτi)cI(tτj)dt respectively. For an M Sequence of Nc points (transmitted over period TC), Rτij = Aτij = 1/Nc for all delays except for τi = τj when Aτij = 1.

For modulation speeds much faster than the phase signal (GHz vs low kHz), the correlation output can be approximated as below, where the detected signal amplitude is βi(t)=αiE˜IN(t)E˜IN(tτi)/2.
x^j(t)=P˜Int(t)cI(tτj)Tc1TctTCtcI(tτj)i=0NSβi(t)[cI(tτi)cosφ˜i(t)+cQ(tτi)sinφ˜i(t)]dt1TctTctβj(t)cosφ^j(t)+βj(t)cI(tτj)cQ(tτj)sinφ˜j(t)+cI(tτj)ijβi(t)[cI(tτi)cosφ˜i(t)+cQ(tτi)sinφ˜i(t)]dt

In the limit of long code and stationary signal, we have the following;
x˜j(t)βj(t)cosφ˜j(t)+βj(t)Rj,jsinφ˜j(t)+ijβi(t)[Aj,icosφ˜i(t)+Rj,isinφ˜i(t)]βj(t)cosφ˜j(t)

Similarly, the imaginary quadrature estimate y^j(t)=1Tc(P˜Int(t)cQ(tτj)) is given by correlation against cQ(tτj);
y˜j(t)βj(t)sinφ˜j(t)+βj(t)Rj,jcosφ˜j(t)+ijβi(t)[Rj,icosφ˜i(t)+Aj,isinφ˜i(t)]βj(t)sinφ˜j(t)

The j and j estimates produced now provide the real and imaginary components required for phase measurement. Since the PRN length sets cross-correlation/coupling magnitude, PRN selection is a compromise between modulation speed, measurement update rate (PRN repetition period) and achievable sensitivity. In practise, the measurement will be perturbed by various noise sources, allowing selection of a reasonable the code length which will allow measurement noise to dominate over any cross-coupling effects.

2.3. Experimental demonstration

With the goal of independently measuring the phase of multiple signals incident upon a common measurement point, the fibre ring interferometer in Fig. 2 was constructed. A ring was selected as it provides a means of performing a consistency check by comparing measurements of subsequent round trips, each of which should experience an equal optical path. This allows demonstration of the high sensitivity and multiplexing capabilities of DEHoI within a simple, compact fibre interferometer. For practical applications, each of these round trip signals replicates the measurement of an independent sensor, utilising the propagation delay dependant detection for the construction of distributed, multi-sensor metrology system.

A 1550nm fibre laser is split between metrology (95%) and LO (5%) branches. The metrology branch is modulated with two M Sequences by a DPMZ modulator. The modulator degrees of freedom are locked through a 5% tap off to a control detector PDC with feedback to the modulator biases [26

26. P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006). [CrossRef]

]. The modulated field is then sent into the fibre ring, where it acquires a calibration phase modulation through the EOM. The fibre ring and coupler were placed in an isolated and acoustically shielded tank, allowing us to probe the sensitivity limit of the sensor. The free ring output port is combined with the LO and the resulting field is measured upon the signal detector PDS.

Table 1. System parameters for the DEHoI implementations with (Calibration) and without (Free-running) the ring EOM.

table-icon
View This Table

All detection and signal processing was performed in real-time on board a high speed Xilinx Virtex 6 FPGA platform hosting a 1.25Gsps ADC, high speed digital output lines for sequence generation and low speed DAC for modulator control. As a compromise between the ADC bandwidth and platform signalling capabilities, two 312.5MHz M Sequence generators were implemented. This results in a 4 times oversampling of each QPSK symbol at the ADC and provides an equivalent PRN symbol length of 0.62m in fibre. This symbol length represents the minimum path length difference between signals that allows separation. The total ring length was initially set at 5m (including EOM), resulting in 8 sequence samples between arrival times. An arm length mismatch of 14m between LO and the fibre ring path length was allowed to introduce additional laser frequency noise coupling and sensitivity. The signal measured at PDS is digitised and fed to 4 independent phasemeter channels. Each phasemeter channel targets the propagation delay corresponding to the first four field components passing the fibre ring; the straight through pass plus three round-trips. Phase estimation for each channel is performed in real-time by a second order digital PLL with 10kHz update rate. Table 2.3 provides a summary of the system parameters implemented.

To demonstrate the delay multiplexed measurement of multiple round passes of the fibre ring, a single phasemeter channel was used to scan the demodulation delay. In this process, the amplitude of the signal is calculated for each delay j from the j and j estimates and is plotted in Fig. 3. High signal amplitude indicates the presence of signal arriving at the particular delay. For the fibre ring, spikes are clearly observed arriving periodically with decaying amplitude as expected for the transient response of the ring.

Fig. 3 Showing the measured impulse response of the interferometer by correlating the measured output against delayed versions of the modulation sequence. The fibre ring round trips can be seen beginning at 41ns with 25.6ns spacing between arrival times.

The signal sensitivity was tested by injection of a calibration signal into the EOM. The 1.2Vpp, 90Hz sinusoidal input results in an expected single pass modulation depth of 266nmpp for an EOM with a nominal modulation response of Vπ = 3.5V for π rad modulation. The result of tracking the first four signals is shown in Fig. 4(a), where a modulation amplitude of 280nmpp per round trip is observed. The observed value agrees with prediction to within 5%, which is less than the EOM manufacturer Vπ uncertainty of 14% or ±0.5V.

Fig. 4 (a) showing the raw time domain displacement measurement of the first 4 arriving signals. A 90 Hz calibration tone with single pass modulation depth of 240nmpp was applied. Each round trip accumulates phase proportional to the number of passes through the modulator. (b) shows the Root Power Spectral Density (RPSD) for isolated measurements of the first, second and third round trips of the fibre ring. Each round trip senses a single pass of the calibration signal along with 50Hz mains pick-up. Harmonics of the calibration frequency indicate a cyclic error of 7nm. The difference between round trips #1 and #2 is shown to demonstrate the limited cancellation resulting from cyclic error.

To quantify noise and tracking performance, three separate measurements of the fibre-ring contribution were isolated from the four multi-pass measurements ie. Round Trip #1 = First Round Trip - Straight Through etc. By expanding the phase arguments (Eq. (3)), we find that all lead, return and prior round trip phase contributions will be common to each measurement (to within a short propagation delay), hence each round trip estimate aims to measure only the phase of the fibre ring itself. Any residual noise in this measurement can be attributed to the metrology system.

After isolating three separate round trips, the Root Power Spectral Density (RPSD) for each component is shown in Fig. 4(b). The calibration tone and 50Hz mains power leakage are equal between each channel. A displacement sensitivity of 2pm/√Hz is reached in the first round trip measurement. Harmonics of the calibration signal indicate a cyclic error of 7nm, estimated by the methods presented in [31

31. T. G. McRae, M. T. L. Hsu, C. H. Freund, D. A. Shaddock, J. Herrmann, and M. B. Gray, “Linearization and minimization of cyclic error with heterodyne laser interferometry,” Opt. Lett. 37, 2448–2450 (2012). [CrossRef]

]. This non-linearity can be attributed to scatter from the calibration EOM and from error introduced in the generation of the modulation constellation.

Since the phase estimation assumes orthogonal I and Q quadratures (Fig. 1(b)), any errors in the QPSK modulation will degrade performance [28

28. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981). [CrossRef]

]. In particular, any relative timing difference between the I and Q modulation sequences incurred during generation or modulation will result in unequal correlation strengths at the receiver. Any such amplitude difference, which may also result from unequal modulation depth (voltage), will skew the ratio between I and Q components and distort the phase measurement through, for example, Eq. (2). These effects will cause a cyclic error ie. a phase dependent phase error of the form φ̃ φ + γsin(φ), γ ≪ 1. Additionally, increased 50Hz mains coupling is also incurred through the DPMZ locking scheme due to feedback gain limitations. As these effects are largely a feature of the DPMZ modulator, future implementations will look to utilise an alternate generation scheme, such as digital synthesis of a 4 level phase modulation through an EOM.

The effect of such cyclic non-linearity is different in closed vs open loop applications. For an open-loop measurement, a nm-scale non-linearity will introduce a systematic phase measurement error which ultimately limits the absolute displacement sensing and common mode suppression between channels to nm scale [29

29. A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990). [CrossRef]

]. This is shown in Fig. 4(b), where suppression of the common modulation between round trips # 1 and # 2 is limited at the nm level. Recent results have demonstrated that the effects of this form of deterministic cyclic error can be calibrated and suppressed [30

30. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992). [CrossRef]

,31

31. T. G. McRae, M. T. L. Hsu, C. H. Freund, D. A. Shaddock, J. Herrmann, and M. B. Gray, “Linearization and minimization of cyclic error with heterodyne laser interferometry,” Opt. Lett. 37, 2448–2450 (2012). [CrossRef]

]. For closed loop applications, which typically involve stabilisation, this non-linearity will manifest as a systematic offset in the nominal stabilisation position, with the stabilisation performance limited by the underlying sensor noise of the system.

To measure the sensor noise, the calibration EOM was removed to reduce backscatter, resulting in a shorter 2m ring. The resulting free running noise performance of the system is shown in Fig. 5. Picometer-scale noise floors were measured for each round trip, with a high-frequency noise floor of {0.8,1,1.5}pm/√Hz achieved for each sensor above 50Hz. The improvement at higher frequencies relative to Fig. 4(b) is attributed to the higher circulating power in the loop after removal of the EOM. Below these frequencies, audio frequency coupling and phase noise from the un-stabilised fibre paths, visible the Straight measurement, limit the ultimate sensitivity. This is re-enforced by the errors visible in the difference between round trips # 1 and # 2 which is shows no further suppression of common signals, indicating that the measured signal is uncorrelated between measurements and most likely the result of non-linearity.

Fig. 5 Showing the free running displacement RPSD. A picometer scale white noise floor reached above 50Hz. Low frequencies are dominated by incomplete cancellation of fibre acoustic noise, as demonstrated by the Straight through pass.

For open loop metrology applications, uncalibrated cyclic error will degrade the suppression of common noise sources that might otherwise be anticipated. To overcome these limitations, we propose an alteration to the readout method which will provide improved sensitivity at low frequencies, where laser frequency and fibre noise tend to dominate, along with further hardware simplifications.

3. Double demodulation

As Fig. 6 shows, this extension will allow further simplification of the metrology system by removing all components which were previously dedicated to the separate LO path. Such a readout will be of natural application where, like the fibre ring, one field component is noticeable larger than any other (such as the promptly reflected field from a cavity or a relative measurement between in-line components). Commonly, this field also defines a useful phase reference point, allowing isolation of the cavity round trip phase with negligible sensitivity to lead or return paths. The use of the promptly reflected field for demodulation of subsequent signals also removes potential cross-talk which can arise due to the difference in field strengths.

Fig. 6 Showing the fibre ring interferometer setup without dedicated Local Oscillator. The Metrology branch is identically prepared and interrogates the same fibre ring as Fig. 2. The largest signal component (lowest loss) results from the straight through pass of the loop coupler. This component may now be used as a LO for the demodulation of the other loop round trip components.

The round-trip power measurement in Fig. 3 clearly indicates that the field bypassing the fibre ring has significant optical power ( 300 μW for this implementation), hence the path length can be inferred directly from the interference between the straight through and round trip fields. Since the interference point between LO (straight pass) and signal paths (i.e.. fibre ring round trips) now occurs at the output port of the ring coupler, there is inherent common mode rejection of lead and return fibre path length noise.

3.1. Phase measurement

With the dedicated LO removed, the power envelope measured upon the detector will be given by the interference between the various metrology signals. With the constant power terms removed, this will be given by;
P˜Int(t)i=0NjiNαiαj2E˜IN(tτi)E˜IN(tτj)[cI(tτi)cI(tτj)cos(φ˜i(t)φ˜j(t))+cQ(tτi)cQ(tτj)cos(φ˜i(t)φ˜j(t))+cI(tτi)cQ(tτj)sin(φ˜i(t)φ˜j(t))cQ(tτi)cI(tτj)sin(φ˜i(t)φ˜j(t))]

The interference between any two fields produces four distinct cross-terms, representing the beat between the I and Q modulations of the two fields. Each of these four interference terms can be isolated in the same manner as the single demodulation case ie. by a correlation against the total modulation sequence formed from the product of two sequences. In practice, this “double demodulation” is trivially implemented by sequential single demodulations prior to averaging.

As such, the relative phase between i’th and jth paths can be detected through demodulation targeted at each of the four sinusoid terms. For clarity, each demodulation component is designated by the demodulation code quadratures X,Y and the path indices i, j as XYi,j. Similarly to the single demodulation case, selection of sufficiently long/uncorrelated codes provides suppression of all other cross-terms in the demodulation and these components have been ignored.
IIx˜i,j(t)=1Tc(P˜Int(t)(CI(tτj)cI(tτj)))βi,j(t)cos(φ˜i(t)φ˜j(t))QQx˜i,j(t)=1Tc(P˜Int(t)(CQ(tτj)cQ(tτj)))βi,j(t)cos(φ˜i(t)φ˜j(t))IQy˜i,j(t)=1Tc(P˜Int(t)(cI(tτj)cQ(tτj)))βi,j(t)sin(φ˜i(t)φ˜j(t))QIy˜i,j(t)=1Tc(P˜Int(t)(cQ(tτj)cI(tτj)))βi,j(t)sin(φ˜i(t)φ˜j(t))

The first two terms represent the individual I and Q quadrature components beating against delayed versions of themselves. These measurements encode the real part of the relative phase. The remaining terms represent the I and Q components beating against each other, which only occurs when there is a relative phase shift between them, hence they encode the imaginary component. With these four estimates we now create a complex phasor i,j(t) which estimates the difference between the fields as;
p˜i,j(t)=(x˜i,jII(t)+x˜i,jQQ(t))+i(y˜i,jIQ(t)y˜i,jQI(t))=2βi,j(t)(cos(φ˜i(t)φ˜j(t))+isin(φ˜i(t)φ˜j(t)))

This estimate can be tracked using equivalent phase measurement algorithms as Section 2.

3.2. Experimental demonstration

To verify operation of the double demodulation measurement scheme, the fibre ring interferometer used in Section 2 was modified to the configuration shown in Fig. 6 by disconnecting the dedicated LO path. In this configuration, the straight through pass of the ring is used as a LO to demodulate the first three round trips of the ring. After applying comparable processing to the single demodulation data, the fibre ring phase was estimated through subtraction of sequential round trip signals. These components have reduced sensitivity to common paths, such as the lead or return fibre, which resulted in an overall quieter signal. The resulting displacement RPSD is shown in Fig. 7 for the three round trips, which shows substantially reduced low frequency acoustic coupling and fibre noise when compared to Fig. 5. The Expected Laser trace shows the sensitivity limitation for a representative laser frequency noise spectrum over a 2m ring length with 103Hz/√Hz at 1Hz. At low frequency, the measurements appear to be limited by the laser frequency noise whilst at high frequencies a 1pm/√Hz white noise floor is measured down to 20Hz. Acoustic noise coupling is visible down to 2Hz. A differential error measured between the first and second round trips was limited to 2pm/√Hz down to 3Hz.

Fig. 7 Shows the displacement RPSD of the double demodulation measurements for the first, second and third round trips of the fibre ring. A picometer scale white noise floor is measured for all channels above above 3 Hz while low frequencies are dominated by laser frequency noise as predicted by the Expected Laser trace.

These results demonstrate a factor of 102 improvement in sensitivity at 1Hz relative to the single demodulation case. This improved performance is a direct consequence of the insensitivity to lead and return fibre noise that is inherent to the double demodulation method and is achieved with a greatly simplified optical configuration. Additionally, the double demodulation method achieves record sensitivity above 2Hz, surpassing the 5pm/√Hz achieved in [13

13. G. de Vine, D. S. Rabeling, B. J. J. Slagmolen, T. T.-Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. a. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express 17, 828–837 (2009). [CrossRef]

] with the use of a stabilised optical cavity.

4. Conclusion

This paper has demonstrated the most sensitive displacement measurements performed using Digitally Enhanced Interferometry techniques. Multiplexed measurements of a fibre ring achieved displacement measurement noise floors of 0.8pm/√Hz and 1pm/√Hz over low audio frequencies for the single and double demodulation DEHoI techniques respectively. Overall system performance was degraded by the presence of a ∼ 7nm cyclic error, providing scope for further improvement and greater sensitivity in future implementations. Both techniques have demonstrated suitable sensitivity for a range of target applications, such as fibre sensing or interferometer control. The simplified optical configuration and ability to multiplex measurements without compromising sensitivity can provide reductions in overall complexity where sensing of multiple degrees-of-freedom are required. Further application-targeted implementations will focus upon decreasing the cyclic error and noise coupling through new DMPZ control schemes.

Acknowledgments

The authors thank Brent Ware, Danielle Wuchenich, Timothy Lam and Jan Burke for useful discussions. This research was supported by the Australian Research Council Discovery Projects funding scheme (project number DP0986003).

References and links

1.

The LIGO Scientific Collaboration, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Rep. Prog. Phys. 72, 076901 (2009).

2.

T. T.-Y. Lam, M. B. Gray, D. A. Shaddock, D. E. McClelland, and J. H. Chow, “Subfrequency noise signal extraction in fiber-optic strain sensors using postprocessing,” Opt. Lett. 37, 2169–2171 (2012). [CrossRef]

3.

B. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. Klipstein, and W. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J Geodesy 10.1007/s00190-012-0566-3 (2012). [CrossRef]

4.

M. Evans, N. Mavalvala, P. Fritschel, R. Bork, B. Bhawal, R. Gustafson, W. Kells, M. Landry, D. Sigg, R. Weiss, S. Whitcomb, and H. Yamamoto, “Lock acquisition of a gravitational-wave interferometer,” Opt. Lett. 27, 598– 600 (2002). [CrossRef]

5.

A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE 7 (2012).

6.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993). [CrossRef]

7.

G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the Laser Interferometer Space Antenna,” Phys. Rev. Lett. 104, 1–4 (2010). [CrossRef]

8.

S. Anza, M. Armano, E. Balaguer, M. Benedetti, C. Boatella, P. Bosetti, D. Bortoluzzi, N. Brandt, C. Braxmaier, M. Caldwell, L. Carbone, A. Cavalleri, A. Ciccolella, I. Cristofolini, M. Cruise, M. Da Lio, K. Danzmann, D. Desiderio, R. Dolesi, N. Dunbar, W. Fichter, C. Garcia, E. Garcia-Berro, A. F. Garcia Marin, R. Gerndt, A. Gianolio, D. Giardini, R. Gruenagel, A. Hammesfahr, G. Heinzel, J. Hough, D. Hoyl, M. Hueller, O. Jennrich, U. Johann, S. Kemble, C. Killow, D. Kolbe, M. Landgraf, A. Lobo, V. Lorizzo, D. Mance, K. Middleton, F. Nappo, M. Nofrarias, G. Racca, J. Ramos, D. Robertson, M. Sallusti, M. Sandford, J. Sanjuan, P. Sarra, A. Selig, D. Shaul, D. Smart, M. Smit, L. Stagnaro, T. Sumner, C. Tirabassi, S. Tobin, S. Vitale, V. Wand, H. Ward, W. J. Weber, and P. Zweifel, “The LTP experiment on the LISA Pathfinder mission,” Classical Quant. Grav. 22, S125 (2005). [CrossRef]

9.

D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett. 32, 3355–3357 (2007). [CrossRef]

10.

S. Golomb, “On the classification of balanced binary sequences of period 2n-1(Corresp.),” IEEE Trans. Inform. Theory 26, 730–732 (1980). [CrossRef]

11.

S. Raghavan, M. Shane, and R. Yowell, “Spread spectrum codes for GPS L5,” in 2006 Aerosp. Conf. Proc. , 1–7 (2006). [CrossRef]

12.

N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation CW lidar,” Appl. Opt. 22, 1382–1386 (1983). [CrossRef]

13.

G. de Vine, D. S. Rabeling, B. J. J. Slagmolen, T. T.-Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. a. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express 17, 828–837 (2009). [CrossRef]

14.

D. M. R. Wuchenich, T. T.-Y. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett. 36, 672–674 (2011). [CrossRef]

15.

O. P. Lay, S. Dubovitsky, D. A. Shaddock, and B. Ware, “Coherent range-gated laser displacement metrology with compact optical head,” Opt. Lett. 32, 2933–2935 (2007). [CrossRef]

16.

K. Dahl, G. Heinzel, B. Willke, K. A. Strain, S. Goler, and K. Danzmann, “Suspension platform interferometer for the AEI 10 m prototype: concept, design and optical layout,” Class. Quantum Grav. 29095024 (2012). [CrossRef]

17.

O. Jennrich, “LISA technology and instrumentation,” Class. Quantum Grav. 26153001 (2009). [CrossRef]

18.

D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc. 873, 654–660 (2006). [CrossRef]

19.

I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser. 154, 012017 (2009). [CrossRef]

20.

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996). [CrossRef]

21.

Y. Surrel, “Fringe analysis,” Top. Appl. Phys. 77/2000, 55–102 (2000). [CrossRef]

22.

E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011). [CrossRef]

23.

G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express 18, 19076–19086 (2010). [CrossRef]

24.

P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 5–7 (1987).

25.

J. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994).

26.

P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett. 18, 2209–2211 (2006). [CrossRef]

27.

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol. 27, 901–914 (2009). [CrossRef]

28.

P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20, 3382–3384 (1981). [CrossRef]

29.

A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng. 12, 7–11 (1990). [CrossRef]

30.

W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng. 14, 91–98 (1992). [CrossRef]

31.

T. G. McRae, M. T. L. Hsu, C. H. Freund, D. A. Shaddock, J. Herrmann, and M. B. Gray, “Linearization and minimization of cyclic error with heterodyne laser interferometry,” Opt. Lett. 37, 2448–2450 (2012). [CrossRef]

OCIS Codes
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology
(120.2920) Instrumentation, measurement, and metrology : Homodyning
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.3940) Instrumentation, measurement, and metrology : Metrology
(120.4640) Instrumentation, measurement, and metrology : Optical instruments
(120.5050) Instrumentation, measurement, and metrology : Phase measurement
(120.5060) Instrumentation, measurement, and metrology : Phase modulation
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: July 25, 2012
Revised Manuscript: August 29, 2012
Manuscript Accepted: August 30, 2012
Published: September 13, 2012

Citation
Andrew J. Sutton, Oliver Gerberding, Gerhard Heinzel, and Daniel A. Shaddock, "Digitally enhanced homodyne interferometry," Opt. Express 20, 22195-22207 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22195


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References

  1. The LIGO Scientific Collaboration, “LIGO: the Laser Interferometer Gravitational-wave Observatory,” Rep. Prog. Phys.72, 076901 (2009).
  2. T. T.-Y. Lam, M. B. Gray, D. A. Shaddock, D. E. McClelland, and J. H. Chow, “Subfrequency noise signal extraction in fiber-optic strain sensors using postprocessing,” Opt. Lett.37, 2169–2171 (2012). [CrossRef]
  3. B. Sheard, G. Heinzel, K. Danzmann, D. A. Shaddock, W. Klipstein, and W. Folkner, “Intersatellite laser ranging instrument for the GRACE follow-on mission,” J Geodesy 10.1007/s00190-012-0566-3 (2012). [CrossRef]
  4. M. Evans, N. Mavalvala, P. Fritschel, R. Bork, B. Bhawal, R. Gustafson, W. Kells, M. Landry, D. Sigg, R. Weiss, S. Whitcomb, and H. Yamamoto, “Lock acquisition of a gravitational-wave interferometer,” Opt. Lett.27, 598– 600 (2002). [CrossRef]
  5. A. Bouchez, B. McLeod, D. Acton, and S. Kanneganti, “The Giant Magellan Telescope phasing system,” Proc. SPIE7 (2012).
  6. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol.4, 907–926 (1993). [CrossRef]
  7. G. de Vine, B. Ware, K. McKenzie, R. E. Spero, W. M. Klipstein, and D. A. Shaddock, “Experimental demonstration of time-delay interferometry for the Laser Interferometer Space Antenna,” Phys. Rev. Lett.104, 1–4 (2010). [CrossRef]
  8. S. Anza, M. Armano, E. Balaguer, M. Benedetti, C. Boatella, P. Bosetti, D. Bortoluzzi, N. Brandt, C. Braxmaier, M. Caldwell, L. Carbone, A. Cavalleri, A. Ciccolella, I. Cristofolini, M. Cruise, M. Da Lio, K. Danzmann, D. Desiderio, R. Dolesi, N. Dunbar, W. Fichter, C. Garcia, E. Garcia-Berro, A. F. Garcia Marin, R. Gerndt, A. Gianolio, D. Giardini, R. Gruenagel, A. Hammesfahr, G. Heinzel, J. Hough, D. Hoyl, M. Hueller, O. Jennrich, U. Johann, S. Kemble, C. Killow, D. Kolbe, M. Landgraf, A. Lobo, V. Lorizzo, D. Mance, K. Middleton, F. Nappo, M. Nofrarias, G. Racca, J. Ramos, D. Robertson, M. Sallusti, M. Sandford, J. Sanjuan, P. Sarra, A. Selig, D. Shaul, D. Smart, M. Smit, L. Stagnaro, T. Sumner, C. Tirabassi, S. Tobin, S. Vitale, V. Wand, H. Ward, W. J. Weber, and P. Zweifel, “The LTP experiment on the LISA Pathfinder mission,” Classical Quant. Grav.22, S125 (2005). [CrossRef]
  9. D. A. Shaddock, “Digitally enhanced heterodyne interferometry,” Opt. Lett.32, 3355–3357 (2007). [CrossRef]
  10. S. Golomb, “On the classification of balanced binary sequences of period 2n-1(Corresp.),” IEEE Trans. Inform. Theory26, 730–732 (1980). [CrossRef]
  11. S. Raghavan, M. Shane, and R. Yowell, “Spread spectrum codes for GPS L5,” in 2006 Aerosp. Conf. Proc., 1–7 (2006). [CrossRef]
  12. N. Takeuchi, N. Sugimoto, H. Baba, and K. Sakurai, “Random modulation CW lidar,” Appl. Opt.22, 1382–1386 (1983). [CrossRef]
  13. G. de Vine, D. S. Rabeling, B. J. J. Slagmolen, T. T.-Y. Lam, S. Chua, D. M. Wuchenich, D. E. McClelland, and D. a. Shaddock, “Picometer level displacement metrology with digitally enhanced heterodyne interferometry,” Opt. Express17, 828–837 (2009). [CrossRef]
  14. D. M. R. Wuchenich, T. T.-Y. Lam, J. H. Chow, D. E. McClelland, and D. A. Shaddock, “Laser frequency noise immunity in multiplexed displacement sensing,” Opt. Lett.36, 672–674 (2011). [CrossRef]
  15. O. P. Lay, S. Dubovitsky, D. A. Shaddock, and B. Ware, “Coherent range-gated laser displacement metrology with compact optical head,” Opt. Lett.32, 2933–2935 (2007). [CrossRef]
  16. K. Dahl, G. Heinzel, B. Willke, K. A. Strain, S. Goler, and K. Danzmann, “Suspension platform interferometer for the AEI 10 m prototype: concept, design and optical layout,” Class. Quantum Grav.29095024 (2012). [CrossRef]
  17. O. Jennrich, “LISA technology and instrumentation,” Class. Quantum Grav.26153001 (2009). [CrossRef]
  18. D. Shaddock, B. Ware, P. Halverson, R. Spero, and B. Klipstein, “Overview of the LISA phasemeter,” “6th International LISA Symposium”, AIP Conf. Proc.873, 654–660 (2006). [CrossRef]
  19. I. Bykov, J. J. E. Delgado, A. F. G. Marín, G. Heinzel, and K. Danzmann, “LISA phasemeter development: advanced prototyping,” J. Phys. Conf. Ser.154, 012017 (2009). [CrossRef]
  20. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt.35, 51–60 (1996). [CrossRef]
  21. Y. Surrel, “Fringe analysis,” Top. Appl. Phys.77/2000, 55–102 (2000). [CrossRef]
  22. E. Hack and J. Burke, “Invited review article: measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum.82, 061101 (2011). [CrossRef]
  23. G. Heinzel, F. Guzmán Cervantes, A. F. García Marin, J. Kullmann, W. Feng, and K. Danzmann, “Deep phase modulation interferometry,” Opt. Express18, 19076–19086 (2010). [CrossRef]
  24. P. Hariharan and B. Oreb, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.5–7 (1987).
  25. J. Proakis and M. Salehi, Communication Systems Engineering (Prentice Hall, 1994).
  26. P. S. Cho, J. B. Khurgin, and I. Shpantzer, “Closed-loop bias control of optical quadrature modulator,” IEEE Photon. Technol. Lett.18, 2209–2211 (2006). [CrossRef]
  27. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol.27, 901–914 (2009). [CrossRef]
  28. P. L. M. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt.20, 3382–3384 (1981). [CrossRef]
  29. A. E. Rosenbluth and N. Bobroff, “Optical sources of non-linearity in heterodyne interferometers,” Precis. Eng.12, 7–11 (1990). [CrossRef]
  30. W. Hou and G. Wilkening, “Investigation and compensation of the nonlinearity of heterodyne interferometers,” Precis. Eng.14, 91–98 (1992). [CrossRef]
  31. T. G. McRae, M. T. L. Hsu, C. H. Freund, D. A. Shaddock, J. Herrmann, and M. B. Gray, “Linearization and minimization of cyclic error with heterodyne laser interferometry,” Opt. Lett.37, 2448–2450 (2012). [CrossRef]

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