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`Simultaneous distributed fibre temperature and strain sensor using microwave coherent
`
`detection of spontaneous Brillouin backscatter
`
`This content has been downloaded from IOPscience. Please scroll down to see the full text.
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`2001 Meas. Sci. Technol. 12 834
`
`(http://iopscience.iop.org/0957-0233/12/7/315)
`
`View the table of contents for this issue, or go to the journal homepage for more
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`This content was downloaded on 18/08/2017 at 18:39
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`You may also be interested in:
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`Attachment 5d: Copy of Document 5 from IOPscience
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`HALLIBURTON, Exh. 1013, p. 0163
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`

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`INSTITUTE OF PHYSICS PUBLISHING
`
`Meas. Sci. Technol. 12 (2001) 834–842
`
`MEASUREMENT SCIENCE AND TECHNOLOGY
`
`www.iop.org/Journals/mt PII: S0957-0233(01)20355-5
`
`Simultaneous distributed fibre
`temperature and strain sensor using
`microwave coherent detection of
`spontaneous Brillouin backscatter
`
`Sally M Maughan, Huai H Kee and Trevor P Newson
`
`Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, UK
`
`E-mail: smm@orc.soton.ac.uk
`
`Received 2 January 2001, accepted for publication 28 February 2001
`
`Abstract
`Simultaneous optical fibre distributed strain and temperature measurements
`have been obtained, by measuring the spontaneous Brillouin intensity and
`frequency shift, using the technique of microwave heterodyne detection.
`The enhanced stability from using a single coherent source combined with
`optical preamplification results in a highly accurate sensor. Using this
`sensor, distributed temperature sensing at 57 km and simultaneous
`distributed strain and temperature sensing at 30 km were achieved, the
`longest reported sensing lengths to date for these measurements. As a
`simultaneous strain and temperature sensor, a strain resolution of 100 µε
`◦
`C were achieved.
`and temperature resolution of 4
`
`Keywords: temperature, strain, coherent distributed fibre sensor, spontaneous
`Brillouin scattering, structural monitoring
`
`1. Introduction
`
`Distributed fibre sensing is currently attracting considerable
`research interest due to its unrivalled capability to provide a
`measured property of interest, such as strain or temperature, as
`a continuous function of linear position along the sensing fibre.
`The ability to measure strain and temperature independently
`over a long range with a high spatial resolution has many
`applications, including those in the power and oil industries
`and also in structural monitoring.
`Several methods have been proposed and demonstrated
`for distributed sensing measurements. One popular method
`is the time-domain technique known as optical time-domain
`reflectometry (OTDR), first demonstrated in 1976 by Barnoski
`and Jensen [1], which utilizes the backscattered Rayleigh
`signal to determine optical loss along a length of fibre. In an
`OTDR system, a pulse of light is transmitted down the fibre and
`the light which is backscattered within the numerical aperture
`of the fibre is detected and measured. The time between
`sending the pulse of light and detecting the backscattered signal
`gives a measure of the distance along the fibre, whilst the
`intensity of the backscattered light provides information about
`the measurand. An alternative, novel method for distributed
`
`sensing, using a frequency-domain approach, was performed
`by Ghafoori-Shiraz and Okoshi [2]. The frequency-domain
`analysis is based on the measurement of a complex baseband
`transfer function, which then provides the amplitude of both
`pump and Stokes wave along a fibre length using a network
`analyser. With the frequency-domain approach, distributed
`temperature and strain measurements have been performed
`with a spatial resolution of 3 m over a 1 kmsensing range [3].
`Systems based on Raman backscatter have proved
`commercially successful as instruments for performing
`distributed temperature measurements, due to the practical
`approach of using conventional silica-based optical fibre as
`the sensing element. However,
`these sensors are unable
`to achieve measurement of distributed strain. As a result,
`another category of sensors, utilizing Brillouin scattering,
`have received much attention. Simultaneous measurement of
`temperature and strain is possible using Brillouin scattering
`since both its frequency shift and power are dependent on both
`of these quantities. Several techniques have been developed for
`obtaining the backscattered Brillouin signal, in order to enable
`the measurement of distributed strain and/or temperature. Both
`stimulated and spontaneous Brillouin scattering regimes for
`distributed sensing have previously been reported.
`
`0957-0233/01/070834+09$30.00 © 2001 IOP Publishing Ltd Printed in the UK
`
`834
`
`Attachment 5d: Copy of Document 5 from IOPscience
`
`HALLIBURTON, Exh. 1013, p. 0164
`
`

`

`In the case of stimulated scattering, access to both
`ends of the sensing fibre, or provision of an end-reflection,
`is required.
`In either the Brillouin-gain or Brillouin-loss
`stimulated scattering mechanism, the measured quantity is
`usually just the Brillouin frequency shift, which is found using
`the interaction between counterpropagating pulsed and CW
`radiation, separated by approximately the Brillouin frequency
`shift [4, 5]. The frequency shift distribution is determined by
`maximizing the increase (or decrease) of the signal at each
`desired point along the sensing fibre; this maximum occurs
`when the frequency difference between the two lasers is equal
`to the Brillouin frequency shift at that point. In this way, it
`is possible to measure either strain or temperature, provided
`that the fibre is either at a constant temperature or strain,
`respectively. Simultaneous measurements using the Brillouin-
`loss technique have been attempted, utilizing the Brillouin-loss
`peak power as well as the frequency shift. However, this was
`only for a sensing length of 50 m of polarization-maintaining
`fibre and required a portion of this length to be kept at a known
`temperature and strain, as a reference [6]. Errors of 178 µε
`◦
`C were measured, for a spatial resolution of 3.5 m,
`and 3.9
`over this 50 m length.
`With spontaneous scattering, access to only one end
`of the fibre is necessary.
`Furthermore, measurement of
`the spontaneous backscattered Brillouin power (normalized
`to the temperature- and strain-insensitive Rayleigh power)
`along with the Brillouin frequency shift, allows simultaneous
`measurement of temperature and strain over tens of kilometres.
`We focus on this type of sensor in this paper. Techniques for
`spontaneous Brillouin backscatter measurement fall broadly
`into two categories: direct detection and coherent (heterodyne)
`detection.
`In direct detection, the Brillouin signal must be
`optically separated from the much larger, elastic, Rayleigh
`component prior to detection. This has been done, for example,
`using Fabry–Perot
`[7, 8] or fibre Mach–Zehnder
`[9, 10]
`interferometers, but these optical filters must necessarily be
`highly stable due to the small frequency difference between
`Brillouin and Rayleigh components (∼11 GHz at 1.5 µm).
`Simultaneous strain and temperature measurements have been
`performed using direct detection, for a sensing length of 15 km
`and a spatial resolution of 10 m, with an RMS temperature error
`◦
`C and an RMS strain error of 290 µε [10].
`of 4
`Coherent detection employs a strong, narrow linewidth,
`optical
`local oscillator
`(OLO) which allows very good
`electrical filtering of the Brillouin component and so a
`much greater tolerance of Rayleigh contamination than direct
`detection. Coherent detection also results in a greater dynamic
`range, since the detector photocurrent, at the beat frequency,
`has only a square root dependence on signal power. Also,
`since the RMS signal photocurrent is much higher than that
`for direct detection, due to optical mixing with the OLO, a
`detector with a higher noise-equivalent power (NEP) may be
`used, for instance a broader-bandwidth detector. To date,
`coherent detection of spontaneous Brillouin backscatter has
`been achieved by arranging for the frequency shift between
`the OLO and sensing pulses to be approximately equal to
`the Brillouin shift, bringing the Brillouin/OLO beat frequency
`within the bandwidth of a conventional heterodyne receiver.
`This frequency shift has previously been attained using a
`Brillouin laser [11], an acousto-optic modulator (AOM) ring
`circuit [12] and an electro-optic modulator (EOM) [13, 14].
`
`Microwave coherent distributed Brillouin sensing
`
`A technique for obtaining distributed spontaneous
`Brillouin backscattered spectra, which employs an 11 GHz
`microwave heterodyne system in conjunction with optical
`preamplification of the signal, has recently been introduced
`[15].
`This sensor combines the advantages of coherent
`detection and spontaneous Brillouin measurement, allowing
`simultaneous single-ended measurement of temperature and
`strain over a long range, but it also exhibits further advantages
`due to the microwave detection frequency.
`In particular,
`since the expected range of Brillouin frequency shift (up
`to ∼500 MHz) lies within a very small percentage of the
`total bandwidth of the detector (∼20 GHz),
`the detector
`gain is almost constant for the entire signal. Also,
`the
`11 GHz detection frequency allows independent observation
`of both Stokes and anti-Stokes spectra using the same optical
`arrangement: the signals are separated in frequency due to the
`shift of the AOM and also filtered optically by a narrow-band
`fibre Bragg grating. Furthermore, since high-frequency optical
`shifting elements are not required, as was the case in previous
`heterodyne systems, the frequency stability of the sensor is
`exceptionally good.
`A brief overview of spontaneous Brillouin scattering
`and the technique for simultaneous strain and temperature
`measurements are provided in section 2. The construction
`and operation of the sensor is described in section 3 and the
`results obtained, including the first simultaneous temperature
`and strain measurements using this technique, are presented in
`section 4. Section 5 contains a summary of our findings.
`
`2. Spontaneous Brillouin scattering for temperature
`and strain measurements
`
`The initial observation of Brillouin scattering in bulk silica
`occurred in 1950 [16]. It has been shown [6, 17–19] that the
`Brillouin backscattered intensity and frequency shift exhibit
`both strain and temperature dependence. If the sensing fibre is
`subjected to both temperature and strain effects it is necessary
`to measure both the Brillouin intensity and frequency shift
`along the sensing fibre to obtain accurate information regarding
`temperature and/or strain.
`Spontaneous Brillouin scattering results when a small
`fraction of the incident light is inelastically scattered by
`thermally excited acoustic waves (acoustic phonons) in the
`optical fibre. A periodic modulation of the dielectric constant
`and hence refractive index of the medium is generated due
`to density variations produced by the acoustic wave. The
`scattered light undergoes a Doppler frequency shift and
`has maximum scattering in the backwards direction. This
`frequency shift is given by
`νB = 2nva
`λp
`
`(1)
`
`where va is the acoustic velocity in the fibre, n is the refractive
`index and λp is the pump wavelength. The exponential decay
`nature of the acoustic waves results in a Lorentzian spectral
`profile.
`is
`the backscattered signal
`The frequency shift of
`approximately three orders of magnitude smaller than for
`Raman scattering, corresponding to the much smaller acoustic
`phonon frequencies involved in Brillouin scattering (∼11 GHz
`
`835
`
`Attachment 5d: Copy of Document 5 from IOPscience
`
`HALLIBURTON, Exh. 1013, p. 0165
`
`

`

`for a pump wavelength in the 1.5 µm wavelength region),
`which makes separation of the Brillouin from the Rayleigh
`signal more difficult.
`The change in Brillouin frequency shift and power due
`to strain and temperature may be represented by the matrix
`equation
`
`S M Maughan et al
`
`(cid:1)
`
` νB
` PB
`
`(cid:1)
`
`(cid:2)
`
`=
`
`(cid:2)(cid:1)
`
`CνB ε CνB T
`CPB ε CPB T
`
` ε
` T
`
`(cid:2)
`
`(2)
`
`where CνB ε and CνB T
`are the strain and temperature
`coefficients for frequency shift and CPB ε and CPB T are
`the coefficients for power variations. The two variables
`of strain and temperature can be resolved by taking the
`inverse of the above equation.
`If the inverse matrix is
`non-singular, i.e. if CνB εCPB T (cid:4)= CνB T CPB ε, then a solution
`exists. For the values of the coefficients obtained in this
`paper, CνB εCPB T /CνB T CPB ε = −19.3 and so simultaneous
`distributed temperature and strain measurement is possible.
`(cid:2)
`(cid:1)
`The inverse equation is given by
`=
`1
`|CνB εCPB T − CPB εCνB T |
`(cid:2)(cid:1)
`(cid:2)
`(cid:1)
`CPB T −CνB T
` νB
`−CPB ε
`CνB ε
` PB
`and the corresponding errors in the derived strain and
`temperature measurements are given by [20]
`
` ε
` T
`×
`
`(3)
`
`|δε| = |CPB T ||δνB| + |CνB T ||δPB|
`|CνB εCPB T − CPB εCνB T |
`|δT | = |CPB ε||δνB| + |CνB ε||δPB|
`|CνB εCPB T − CPB εCνB T | .
`
`(4)
`
`(5)
`
`3. Experimental arrangement
`
`The experimental configuration for the microwave heterodyne
`spontaneous Brillouin-based fibre sensor is shown in figure 1.
`
`3.1. The source
`
`Excellent frequency stability was ensured by deriving both the
`sensing pulses and the local oscillator from the same seed laser:
`a 100 µW continuous wave, fibre-pigtailed laser, tunable from
`∼1520 to 1560 nm. The source itself was designed to be of dual
`nature. In one setting, used for the Brillouin measurements,
`the source was narrowband, with the linewidth of the seed
`laser (1 MHz); this was achieved with the fibre optic switch
`in position 1, the seed laser being amplified by the erbium-
`doped fibre amplifier, EDFA1.
`In the second setting, with
`the switch in position 2, the source was broadband (∼6 nm)
`and partially polarized, due to ASE feedback into EDFA1
`from a broadband reflecting mirror via a pigtailed polarizer.
`The partial polarization of the ASE was necessary to aid its
`subsequent passage through the polarization-sensitive electro-
`optic modulator (EOM). The source output was ∼12 mW in
`either setting. Radiation from the source was split by a 3 dB
`fibre coupler into pulse and local oscillator arms.
`
`836
`
`3.2. Pulse formation
`
`Pulses were initially formed by a 110 MHz, downshifting,
`fibre-pigtailed AOM before amplification by EDFA2 to give
`pulses up to 4.5 W peak power at 150 ns pulse width. An
`electro-optic modulator (EOM), of 5 dB insertion loss, was
`then used to gate the pulses in order to attenuate the throughput
`of ASE between pulses. The pulses were then passed through a
`PZT-based polarization scrambler (insertion loss 3 dB) to help
`reduce polarization noise observed on the signal. A second
`polarization scrambler, also with 3 dB insertion loss, was
`placed in the local oscillator arm to further reduce the noise.
`Using this arrangement, pulses of up to 350 mW could be
`launched down the 30 km of sensing fibre using a 3 dBcoupler.
`In these experiments, pulses of between 150 and 160 mW and
`150 and 200 ns were chosen, since spectral distortion occurs
`for much higher powers. A 95/5 fibre coupler was used as
`a tap for 5% of the backscattered signal, enabling separate
`direct detection of the Rayleigh trace, when operating in the
`broadband mode.
`
`3.3. Brillouin preamplification
`
`In narrowband mode, due to the low sensitivity of the detection
`system,
`the backscattered traces were preamplified using
`EDFA3 (small signal gain of 26.4 dB). Both the Rayleigh
`backscatter and the ASE from EDFA3 were then filtered
`out by reflection from an in-fibre Bragg grating (FBG)
`(reflectivity = 99.4%, λ = 1533.11 nm, λ = 0.12 nm),
`via a circulator. Either the anti-Stokes or Stokes signals
`could be observed by tuning the narrowband source to
`1533.20 nm or 1533.02 nm respectively. Contact with the
`heavy metal optical bench and the use of air conditioning
`both increased the stability of the grating and so no thermal
`drift problems were encountered. Since a typical FBG central
`−1 and
`wavelength temperature sensitivity is 10–15 pm K
`the grating had a flat transmission peak of width 50 pm,
`ambient temperature changes of a degree or two were tolerable.
`Use of a thermally compensated grating package would have
`reduced this problem still further. The attenuation of the
`Rayleigh component rendered negligible its behaviour as a
`weak secondary oscillator. This is the principal method by
`which the Rayleigh can affect the Brillouin signal and is much
`less significant than an equivalent amount of contaminating
`Rayleigh power in direct detection, since it is the ratio of OLO
`power to Rayleigh power which is important, not the ratio of
`Brillouin power to Rayleigh power.
`
`3.4. Detection system
`
`The amplified, filtered backscatter was mixed with the local
`oscillator via a 3 dBcoupler and then detected using a
`−1). The
`20 GHz optical detector (responsivity of 35 V W
`Brillouin/OLO beat spectra were observed using a 26.5 GHz
`RF spectrum analyser, set in zero span mode. In this mode,
`a time-domain trace is obtained for the selected RF beat
`frequency. The maximum available RF resolution bandwidth,
`of 5 MHz, was selected, allowing a spatial resolution of 20 m to
`be achieved. The spectra were built up by taking time-domain
`backscatter traces for a series of beat frequencies, covering
`the expected range of Brillouin shifts. Since the required
`
`Attachment 5d: Copy of Document 5 from IOPscience
`
`HALLIBURTON, Exh. 1013, p. 0166
`
`

`

`Microwave coherent distributed Brillouin sensing
`
`Figure 1. Experimental arrangement of the microwave heterodyne spontaneous Brillouin-based temperature and strain sensor.
`PS = polarization scrambler, AOM = acousto-optic modulator, EOM = electro-optic modulator, EDFA = erbium-doped fibre amplifier,
`FBG = fibre Bragg grating.
`
`Brillouin power was necessarily proportional to RF power, but
`the recorded traces were proportional to RF voltage, squaring
`of the data was necessary. Any dc interpulse level was then
`subtracted and processing of the spectra was undertaken.
`
`3.5. Spectrum processing
`
`After each set of spectra was obtained, the frequency shift
`and power of the Brillouin backscatter was determined for
`each point of interest along the fibre. This was done by
`fitting each individual spectrum to a Lorentzian curve, since
`the spontaneous Brillouin line is known to be of this shape.
`The Levenberg–Marquardt nonlinear least squares algorithm
`was used for this purpose [21]. The total power, being
`proportional to the area under the curve, was then found. For
`the Lorentzian spectral profile, total power is proportional to
`peak power multiplied by linewidth. At certain points along the
`sensing fibre, where the frequency shift changes significantly
`over a distance smaller than the spatial resolution, a single
`Lorentzian curve is insufficient to determine the backscatter
`characteristics. To overcome these visible transitional hiccups,
`a double or even triple Lorentzian was fitted.
`
`4. Distributed sensing results
`
`Firstly, examples of the Lorentzian curve fitting are presented,
`to show the validity of the process. After this, distributed
`results for a 57 km sensing length are discussed, revealing
`the range limit for this system as a simultaneous temperature
`and strain sensor. A calibration of the dependence of
`both frequency shift and backscattered power on temperature
`was undertaken at this stage and the coefficients compared
`to previously measured values.
`Finally,
`simultaneous
`measurement of temperature and strain are discussed for a
`30 km sensing fibre.
`
`4.1. Lorentzian curve fitting
`
`A sample set of distributed anti-Stokes Brillouin spectra is
`shown in figure 2(a) for a 3.5 km section, located 25 km down
`◦
`the sensing fibre. A 500 m heated portion (at 65
`C) is clearly
`
`= 1
`N
`
`(6)
`
`χ 2
`
`N
`
`visible due to its frequency shift from the unheated regions.
`Figure 2(b) shows a single spectrum from this 3.5 km section
`and its corresponding fitted Lorentzian curve. To estimate
`the goodness of fit, the value χ 2/N was calculated, which is
`N(cid:3)
`defined by
`(yi − f (xi ))2
`σ 2
`i=1
`i
`for a data set of N points, (xi...N , yi...N ), with standard errors in
`y of σi...N , being modelled to a function f (x). χ 2/N should
`be roughly equal to unity for a good fit with the expected
`noise characteristics, with a closer fit being indicated by a
`lower value. To obtain an estimate in this case, the noise
`on each point was assumed to be identical and dominated by
`electrical noise, which was calculated as the standard deviation
`of the inter-pulse, flat, spectrum. The measured value of
`χ 2/N for figure 2(b) is 0.82, validating the choice of spectral
`profile. Examples of double and triple curve fitting results
`at ∼31 km down the sensing fibre are shown in figure 3.
`χ 2/N values for these two curves, measured in an identical
`manner as before, are 1.24 and 0.86, again showing agreement
`with the model. Of course, the inclusion of any additional
`noise sources would decrease χ 2/N, for any given measured
`spectrum, since the standard error used in equation (6) would
`be larger. Automation of the processing may be achieved by
`firstly fitting to each spectrum a single Lorentzian curve; if
`χ 2/N is high, however, a double peak may then be tried, or a
`triple peak, and so on, until a good fit is obtained.
`
`4.2. Measurements over a 57 km sensing fibre
`
`In order to gauge the potential performance of the sensor,
`distributed anti-Stokes Brillouin spectra were obtained over a
`57 km sensing fibre, the longest yet presented using single-
`ended detection of spontaneous Brillouin backscatter. The
`frequency shift and backscattered power measurements are
`shown in figure 4 for this fibre. These were obtained by taking
`a series of 25 different backscatter traces, each separated by
`5 MHz, starting at 10.84 GHz; each trace was averaged 4096
`times. The frequency measurements highlight the boundaries
`between different fibre sections, with the sharp troughs being
`
`837
`
`Attachment 5d: Copy of Document 5 from IOPscience
`
`HALLIBURTON, Exh. 1013, p. 0167
`
`

`

`S M Maughan et al
`
`Figure 3. Example (a) double and (b) triple fitted Lorentzian
`curves. These are both for points ∼31 km along the sensing fibre.
`
`Figure 2. (a) Example distributed anti-Stokes Brillouin spectrum at
`∼25 km distance along the sensing fibre. A 500 m heated section at
`◦
`65
`C is clearly visible. (b) Sample fitted Lorentzian curve (solid
`line) and the original data points (circles) for a single point at 24 km
`along the fibre.
`
`attributed to slack regions between wound drums. The sensing
`fibre comprises five separate fibre lengths, 17 500 m, 17 500 m,
`17 500 m, 500 m and 4000 m, with the 500 m portion being
`placed in an oven and unwound from the drum to ensure
`the absence of strain and the rest of the fibre kept at the
`◦
`C. The frequency measurements show
`room temperature of 22
`clearly, at ∼53 km along the fibre, the shift due to the 500 m
`◦
`C. Each unheated fibre section has a
`heated section, held at 40
`different frequency shift, which may arise from differences in
`winding tension or intrinsic fibre properties (refractive index
`or acoustic velocity). The power measurements show the
`expected exponential decrease with fibre length, agreeing with
`the predicted attenuation coefficient (∼0.4 dB km
`−1 double
`pass at 1.53 µm).
`The RMS noise in both the frequency shift and power
`traces were found over 2 km sections (10 data points) located
`at several positions along the fibre. The power values were
`found after first normalizing the observed trace to a fitted
`exponential function, one for each separate section of fibre.
`
`838
`
`Figure 4. Distributed anti-Stokes Brillouin measurements for an
`entire 57 km fibre length. Both frequency shift and power traces are
`shown.
`
`This information is plotted in figure 5(a) for the frequency
`shift and figure 5(b) for the power. The noise levels increase to
`1.3 MHz and 5.8% at 50 km, corresponding to ∼1.2
`◦
`C/28 µε
`and ∼16
`◦
`C/6500 µε respectively. The power trace is clearly
`too noisy to allow a useful simultaneous sensor at this distance.
`Figure 5(b) indicates that a 1.5% RMS error would occur
`at 30 km, which brings the temperature error due to the
`◦
`C. The RMS power
`power measurement down to less than 5
`error remains at an approximately constant value of 0.7–0.8%
`for the first 20 km of the sensing fibre, over which the
`backscattered power has decreased by ∼8 dB. This indicates
`that polarization noise, which may be expected to have a
`constant percentage value, has not been fully eliminated and
`so improved scrambling is necessary, for optimum resolution.
`For an unstrained fibre, the frequency shift gives a direct
`measurement of temperature and, with this application in mind,
`
`Attachment 5d: Copy of Document 5 from IOPscience
`
`HALLIBURTON, Exh. 1013, p. 0168
`
`

`

`Microwave coherent distributed Brillouin sensing
`
`Figure 5. RMS error, taken over a 2 kmwindow, in both
`(a) frequency shift and (b) power for the 57 km sensing fibre, plotted
`at several points along the fibre length.
`
`calibrated temperature measurements were obtained for the
`heated section at 53 km. The RMS noise was calculated to
`be less than 2 MHz over the heated portion for each oven
`temperature; the traces are shown in figure 6(a). The expected
`linear relationship between frequency shift and temperature
`is clearly visible in figure 6(b), with the coefficient being
`1.07 ± 0.06 MHz K
`−1, agreeing in magnitude with other
`sources [22, 23].
`
`4.3. Power measurements over a 27 km sensing fibre
`
`Power measurements are more complicated to obtain than
`frequency shift measurements.
`Initially, the sensing length
`was merely reduced to 27.4 km (four sections of 17 500 m,
`8900 m, 500 m and 500 m, with the third section being
`heated). The same technique as before was applied (this time
`for 25 frequencies separated by 5 MHz, starting at 10.85 GHz,
`each averaged 12 288 times) and a single Lorentzian was fitted
`for each point along the fibre. Discontinuities in temperature,
`however, resulted in sharp spikes in the recorded power,
`either side of the heated section. This is clearly visible in
`figure 7(a), which shows how the power measurements, at
`26.5 km, depend on temperature. Ignoring the anomalies at
`either end of the heated section, another linear relationship is
`revealed and is shown in figure 7(b). The coefficient relating
`the percentage change in power to temperature was calculated
`as 0.36±0.04% K
`−1, again agreeing with other sources [7, 22].
`The artificial peaks may be removed, however, by fitting a
`double Lorentzian curve at the transitional points, as in figure 8.
`The RMS error in temperature was found to be less than 3.4 K
`(equivalent to 1.2% power error) at the heated section.
`
`Figure 6. Variation of Brillouin