Noise compensation in a Fabry-Perot-based displacement sensor operating at picometer-level resolution

An approach for compensating the influence of the interrogator noises on the readings of the extrinsic Fabry-Perot interferometric displacement sensor is proposed. With the use of a reference interferometer and special processing, the sensor resolution was about twofold improved and comprised 12-25 pm instead of 15-46 pm for a wide range of EFPI baselines.


Introduction
During the last two decades a great progress in manufacture and implementation of the fiber optic sensors based on the extrinsic Fabry-Perot interferometers (EFPI) [1] has been achieved by the academic institutions and commercial companies. Such sensors demonstrate a great dynamic measurement range and a high resolution [2], [3]. Sensors of a great variety of physical quantities have been designed and implemented. The most commonly used EFPI baseline demodulation approaches are the white-light interferometry, using a scanning readout interferometer [4], [5] and wavelengthdomain interferometry, in which the measurement and further processing of the interrogated interferometer spectral function is used to measure the interferometer OPD [6]- [9]. The best attained baseline resolution was 14-15 picometers [10], estimated as doubled standard deviation of the measured baseline fluctuations. The analysis of the noise sources and the light propagation inside the EFPI cavity was made in a paper [10], dedicated to resolution limits analysis for single EFPI sensors.
An idea of compensating hardware noises in interferometric sensing systems had been proposed long ago, mainly dedicated to reducing the laser frequency noise influence. In a paper [11] the idea of noise reduction with a compensating interferometer was realized for extrinsic Fabry-Perot sensors, interrogated by means of the wavelength-domain interferometry. As a result, the sensor resolution was tenfold improved (from 700 nm to 70 nm).
However, the approach proposed in [11] is suitable only for suppression of the fluctuations, induced by environmental influences and isn't able to suppress OPD noises, induced by fluctuations in the acquisition hardware. The goal of the current paper is therefore to develop an approach of OPD resolution enhancement by means of reducing the influence of hardware noises on the OPD readings. This will provide an ability to either use cheaper interrogation units with poorer characteristics or to improve the resolution of high end displacement sensors.
Throughout this paper we consider the case of registering the spectrum of the light reflected from the sensor, which is the most common case for both single-sensor and multiplexed systems. The spectral function of a low-finesse Fabry-Perot interferometer is a sum of quasi-static and quasiharmonic components, and is expressed as SFP(L, λ) = S0(L, λ) + SM•S(L, λ), where [10]: (3) and an approximation of Gaussian profile was applied to the fiber mode and the beam inside the interferometer. η(L) is a coupling coefficient of a light beam, irradiated by a fiber mode and travelled a distance 2L back to a fiber mode [10]; L is the interferometer baseline; w0 is an effective radius of a mode at the output of the first fiber; λ is the light wavelength; n is the refractive index of the media between the mirrors; zR=πnw0 2 /λ -Rayleigh length of the beam; the argument additive γR(L, λ) contains a phase shift ψR, induced by the light diffraction inside the cavity, and a phase φR, induced by the mirrors (typically for dielectric mirrors φR=π). The equations above are valid for the case of parallel mirrors.
One of the most attractive approaches for estimating the baseline L from the registered spectrum is to approximate its variable component S'(λ) with analytical expression (2) by means of least-squares fitting. Such fitting returns the global minimum of the residual norm, given by expression where S'i=S'(λi), Si(L)=S(λi, L), λi = λ0 + i•Δ, Δ is the step between the spectral points, i = -(M-1)/2,…(M-1)/2, M is the number of points in digitized spectrum (for the current notation M must be odd, which corresponds to the utilized interrogator and the performed simulations).

Noise mechanisms
An extensive study of single EFPI displacement sensors resolution limits with wavelength-scanning interrogation was done in [10]. It was shown that the main noise sources are: 1. Absolute wavelength scale shift Δλ0, determined by fluctuations of the triggering of the scanning start, σΔλ=stdev{Δλ0}.
2. Jitter of the wavelength points δλi, caused by the fluctuations of the signal sampling moments, σδλ=stdev{δλ}.
These mechanisms will result in distortion of the registered interferometer spectral function S'(λ). Therefore, the spectrum approximation procedure gives an erroneous result, denoted throughout the paper as La. When considering a vector of consequently measured baseline values, its standard deviation σLa can be calculated. Generally, σLa is used as a quantitative characteristic of a sensor resolution, which is approximated as 2•σLa.
The first mechanism provides the shift of the measured interferometer spectrum, inherently shifting the displacement sensor readings as follows δL ≈ -Δλ0•L0/λ0. (5) The jitter of spectral points during interrogation produces the distortion of the measured spectral function S'(λ). This distortion can be interpreted as an additive noise with some variance. The resulting signal-to-noise ratio SNRJIT can easily be estimated by simple trigonometric derivations [10]: Considering the third mechanism, one has to take into account that generally the noise variance can depend on the mean optical power PI, incident to the photodetector (shot noise level and laser intensity noise influence are strongly related to the mean power level). The dependency can be adequately approximated by a power function as follows σs = aPI b , (7) which, for an EFPI, produces the signal-to-noise ratio of the spectrum, written as follows P0 is the optical power irradiated by the light source; R1,2 * = R1,2•η are effective mirrors reflectivities, taking into account light losses due to divergence of a non-guided beam inside the cavity. The parameters a and b must be obtained explicitly for a given experimental setup.
The influence of the additive noises on the standard deviation of the baseline measurement can be found either by numeric simulation, or analytically using the Cramer-Rao bound [12]. It can be shown that for the approximation approach [3] the corresponding relation can be written as σL(SNR) = C•SNR -1/2 , (9) where C is within the interval (9 ÷ 11)•10 -4 μm -1 for different parameters of the baseline measurement approach [3].
Finally, the expression for the baseline standard deviation can be obtained by combining expressions (5) and (9) and taking into account the variance summation rule: Substituting (2), (6) and (8) to (10), one will be able to obtain the final explicit expression, which isn't done due to excessive bulkiness.

Baseline noises statistics
Throughout the paper the following noise compensation scheme will be assumed: two interferometers with similar parameters are used for the measurement, one of them is exposed to the target perturbation (which is the actual measurand) and will be referred to as sensing or signal interferometer, and the second one is isolated from any environmental changes and will be referred to as reference interferometer. The interferometer baselines will be denoted LS and LR, respectively. Both these interferometers will be assumed to be interrogated by the same tunable laser, while their spectral functions will be registered by independent (although, similar) photodetectors.
Considering the task of noise compensation, one needs to determine the likelihood of the two interferometers' measured baselines fluctuations, which can be done by means of correlation coefficient. Therefore, in order to study the ability of noise cancellation, one needs to study the behavior of the correlation coefficient CBF between the registered baseline values of reference and sensing interferometers with respect to the mechanisms, mentioned in the previous section.
As can be seen from expression (5), the absolute shift of the wavelength scale produces an error, proportional to the baseline value, therefore, its influence will be correlated for the signal and reference interferometers with arbitrarily different baselines.
Considering the impact of individual spectral points jitter, one has to take into account that it is indirect in nature: jitter itself produces equivalent additive noises, which, in turn, affect the accuracy of the approximation procedure [3]. If the sensing and the reference interferometers have equal baselines LS ≈ LR, the shapes of their spectral functions will be the same, resulting in identical noise patterns, produced by the jitter. This will provide equal baseline measurement errors for the both interferometers and CBF = 1. Recall the properties of the spectral function S'(λi), particularly that it is quasi-harmonic with period close to λ0/2, one can expect secondary maximums of correlation in cases |LS -LR| =λ0/2 and |LS -LR| =λ0/4 (when interferometers spectral functions are nearly inversed). On the other hand, for arbitrarily different baselines LS and LR, the spectral functions will be uncorrelated, producing sufficiently different noise patterns, which will result in uncorrelated baseline noises. The dependency of the produced baseline noises correlation on the difference of interferometer baselines can be found by means of the numeric calculations.
Considering the additive noises (the third mechanism), one needs to take into account their two main sources -laser intensity noises, equal for the both interferometers and the photodetector noises, which are produced by different devices, and hence, are uncorrelated, producing uncorrelated errors. Laser intensity noises, being the same for the measurement and reference interferometers, will produce identical errors in case of close baseline values LS ≈ LR and |LS -LR| =λ0/2. In case of |LS -LR| =λ0/4 the influence of laser intensity noises will be inverse for signal and reference interferometers and will produce anti-correlation CBF(λ0/4) = -1. For arbitrarily different baselines the baseline fluctuations will be uncorrelated, CBF = 0. Dependency CBF(LS -LR) for a particular setup will depend on the relation of the above mentioned noise sources.
In order to study the exact dependency of the signal and reference interferometers' baseline noises correlation CBF on the baseline difference, a numeric simulation has been performed. Two spectral functions S(LS,R, λ) were calculated according to (1)  From this dependence it can be predicted that for baselines difference less than 50 nm the correlation is still enough for noise compensation.

Experimental noise compensation
In order to support the theoretical results, an experimental study of EFPI displacement sensor resolution was carried out. Spectral measurements were performed using the optical sensor interrogator NI PXIe 4844, utilizing a tunable laser with SMF-28 single-mode fiber output.  The sensing and reference interferometers were formed by the two ends of SMF-28 fiber packaged with PC connectors, fixed in a standard mating sleeve. The air gaps LS and LR between the fiber ends was varied from ~100 μm up to ~800 μm with steps ~100 μm by the use of Standa 7TF2 translation stages. The experimental setup is schematically shown in figure 2.
The aim of the first performed experiment was to verify the correlation properties of the baseline fluctuations, predicted by means of numeric modeling. As in the modeling, the baselines of the both interferometers were set to ~ 200 μm, the signal interferometer baseline was fixed, while the reference interferometer baseline was scanned with step ~100 nm. In figure 1 the experimental relation CBF(δL) (dots) is compared with the simulated dependency CBF(δL) (solid line). Not exact correspondence of simulated and experimental dependencies can be due to some discrepancy of experimental setup parameters and those implied in the simulation. Also, a lower level of correlation is observed in experiment, therefore, limiting the admissible baseline discrepancy to somewhat 10 nm. After that the noise compensation possibilities were tested. In order to compensate the parasitic baseline fluctuations, the baselines of both the signal and reference interferometers LS and LR were measured, after that the deviations of the reference interferometer baseline from the initial value were subtracted from the sensing interferometer baseline values: The baselines of both interferometers were set nearly equal with accuracy better than 1 nm. For each baseline value the measurements were performed for about 5 minutes, resulting in 300 measured points, with respect to them the standard deviation was calculated. Then the baselines were changed and measurements were repeated for another LS and LR values. The dependencies of standard deviations of a single sensor readings LS and noise-suppressed double sensor readings LSC on the baseline value are shown in figure 3. Estimation of the measured baseline standard deviation calculated according to (10) is also presented as a solid curve for reference.  As can be seen in figure 3, in the performed experiment the effort of the noise compensation exceeded 1.5 for baselines greater than 400 μm and reached 2 for LS ≈ LR ≈ 700 μm.

Conclusion
In the current paper an original approach for eliminating the influence of the interrogating unit noises on the measured value of interferometer baseline is proposed. Parasitic baseline fluctuations, induced by unideal interrogator operation are estimated by means of additional reference interferometer, isolated from the environment, which had the same parameters as the sensing interferometer. After that the fluctuations of the reference interferometer measured baseline were subtracted from the measured baseline of the signal interferometer. As a result, the standard deviation of the signal interferometer measured baseline value was about twofold decreased.