INOM
EXAMENSARBETE TEKNISK FYSIK, AVANCERAD NIVÅ, 30 HP
STOCKHOLM SVERIGE 2020,
Distributed Temperature Sensing Using Phase-Sensitive Optical Time Domain Reflectometry
SIMON EK
KTH
SKOLAN FÖR TEKNIKVETENSKAP
Degree Project in Engineering Physics, Master Level SK202X
Distributed Temperature Sensing Using
Phase-Sensitive Optical Time Domain Reflectometry
Simon Ek
Supervisor:
Kenny Hey Tow
Examiner:
Fredrik Laurell
September 30, 2020
Abstract
This thesis explores and evaluates the temperature measuring capabilities of a phase- sensitive optical time-domain reflectometer (φ-OTDR), which exploits Rayleigh backscat- tering in normal single mode optical fibers. The device is constructed and its setup explained, and a protocol for making temperature measurements with it is developed.
Performance tests are made and the device is shown to achieve fully distributed temper- ature measurements on fibers hundreds of meters in length with a spatial resolution of 1 m and a temperature resolution of 0.1 K. In addition, the capabilities of the device to measure normal strain in the measurement fiber are tested using the same approach, albeit with less success. The device is capable of very precise measurements, making it very sensitive to the environmental conditions around the measuring fiber but also susceptible to disturbances. Some discussion is had on how to avoid or deal with these disturbances. Furthermore, the technique is shown to be able to run in conjunction with other φ-OTDR measurement techniques from the same device simultaneously.
Sammanfattning
Det h¨ar examensarbetet utforskar och utv¨arderar f¨orm˚agorna att m¨ata temperatur hos en fas-k¨anslig optisk tidsdom¨an-reflektometer (φ-OTDR), som utnyttjar bak˚atriktad Rayleigh-spridning i vanliga optiska singelmodfibrer. Anordningen konstrueras och dess komponentstruktur f¨orklaras, och ett protokoll tas fram f¨or att utf¨ora m¨atningar med den. Prestandatester utf¨ors och anordningen visas kapabel att g¨ora fullt distribuerade temperaturm¨atningar l¨angs hundratals meter l˚anga fibrer, med en rymdsuppl¨osning p˚a 1 m och en temperaturuppl¨osning p˚a 0.1 K. Dessutom testas f¨orm˚agan att m¨ata nor- malt¨ojning hos testfibern med samma metod, dock med mindre framg˚ang. Anordningen
¨ar v¨aldigt k¨anslig f¨or f¨orh˚allandena i omgivningen runt m¨atningsfibern, vilket g¨or den kapabel till m¨atningar med mycket h¨og precision, men ocks˚a mottaglig f¨or st¨orningar.
Lite diskussion h˚alls kring hur dessa st¨orningar kan undvikas eller hanteras. Vidare visas att m¨atningstekniken kan k¨oras samtidigt som andra φ-OTDR-baserade tekniker fr˚an samma anordning.
Acknowledgements
First and foremost I extend my gratitude to my supervisor Kenny Hey Tow for being a great help in all possible aspects of the project, including practical, theoretical, admin- istrative and more.
Secondly, I want to thank Prof. Fredrik Laurell for taking his time to be my exam- iner and Lars-Gunnar Andersson for proofreading the text and making sure the project was moving forward.
I also give my appreciation to all the people at the fiber optics department at RISE in Kista for providing a warm, welcoming atmosphere and for aiding me in a wide vari- ety of situations. In particular I would like to mention Carolina Franciscangelis for her technical solutions and expertise in the role as secondary supervisor, and my good friend Frans Forsberg for putting me in touch with RISE as well as providing assistance and quality company during lab work.
Last but not least, I thank the rest of my friends and family for all the emotional support and opportunities for recreation and relaxation they have given me.
Acronyms
• φ-OTDR — Phase Optical Time-Domain Reflectometer
• APD — Avalanche Photodiode
• ASE — Amplified Spontaneous Emission
• CW — Continuous Wave
• EDFA — Erbium-Doped Fiber Amplifier
• FBG — Fiber Bragg Grating
• FUT — Fiber Under Test
• NA — Numerical Aperture
• OTDR — Optical Time-Domain Reflectometer
• SMF — Single Mode Fiber
• SOA — Semiconductor Optical Amplifier
• VA — Variable Attenuator
• WG — Waveform Generator
Contents
1 Introduction 4
2 The phase-sensitive optical time domain reflectometer 6
2.1 Rayleigh scattering in optical fibers . . . 6
2.2 The principles of an OTDR . . . 7
2.3 The principles and measurement capabilities of φ-OTDRs . . . 9
2.3.1 ∆n and restorability . . . 10
2.4 How to measure ∆n . . . 11
2.4.1 The correlation method . . . 11
2.5 The temperature dependence of n . . . 14
2.6 The strain dependence of n . . . 14
3 Experimental validation 16 3.1 The setup . . . 16
3.1.1 The measurement protocol . . . 17
3.1.2 Post processing . . . 18
3.2 Characterization of temperature sensitivity . . . 20
3.3 Measuring temperature . . . 23
3.4 Characterization of strain sensitivity . . . 25
3.5 Simultaneous temperature & vibration sensing . . . 27
3.6 Making better correlation measurements . . . 29
4 Discussion 33 4.1 Perspective & future work . . . 33
4.2 Summary & conclusion . . . 35
Bibliography 36
Chapter 1 Introduction
Over the last decades, optical fibers have become a common sensing element in devices designed to make distributed measurements. This is partly due the fact that extending the sensing range of these types of devices by several kilometers requires little effort and low cost as the fibers themselves are in general very simple to manufacture and replace.
Other advantages of using fibers over installing arrays of conventional point sensors in- clude their high versatility, such as their small size and low mass, and their resistance to unforgiving environments, such as strong electric fields and high temperatures.
One distributed sensing device that has become prolific is the Optical Time-Domain Reflectometer (OTDR). The OTDR measures the intensity of Rayleigh backscatter from different points along a fiber to infer information about optical attenuation in the fiber itself, which is useful in the telecommunications industry. By adjusting the light source in the device, it is possible to make it phase-sensitive and thereby making it highly responsive to small and local disturbances. This phase-OTDR (or φ-OTDR) was first used in intrusion detection [1] which has eventually lead to more refined acoustic vibration sensing techniques for structural health monitoring among other things [4], but using it for temperature measurements has also been explored since then [2, 3].
Temperature sensing with fibers is not completely new. For example, fiber Bragg gratings (FBGs) are a common way to make temperature measurements with high pre- cision. These are point sensors however, and lack the advantages of fully distributed measurement devices. Another method is based on utilizing Raman scattering, which is distributed. This method is far from as precise as the Rayleigh-based φ-OTDR method can be, though [2].
In this degree project, a φ-OTDR is constructed and its temperature measuring ca- pabilities are investigated and validated. The objective is develop a fully distributed temperature measurement system at RISE, capable of monitoring kilometer long fibers with a 1 m spatial resolution. This is done by constructing the system setup, developing a protocol on how to carry out measurements, and making tests on the φ-OTDR’s per- formance. As the measurements do not yield straightforward values of the parameter of interest, but rather a larger collection of data from which it must be deciphered, code is also written to process and analyze the acquired data.
The work is largely based on a method first proposed and tested by Xin Lu [3], which includes mapping out backscattering intensity profiles and cross-correlating them. Effec- tively, the method sweeps the laser wavelength in order to measure local differences over time in the refractive index of the fiber, which depend on changes in the environment sur-
rounding it. This means that the same method can be used to measure any quantity that affects the refractive index, for example mechanical strain in the fiber material, which is also tested. Since the measurements give differences in the refractive index rather than absolute values, it also means that the temperature measurements are given in differ- ences. The approach requires as such a reference measurement at a known temperature distribution to be able to measure absolute temperature.
While the work is not as thorough as Lu’s, it differs in a few aspects. Firstly, the scale of the temperature changes are quite different. While Lu focuses on the impressive temperature resolution on the ∼ 10−3 K scale, this work establishes that the method also works with a laser wavelength step size 10 times as large. This is beneficial for larger temperature shifts (>1 K) as it keeps down the total amount of steps needed, albeit with diminished performance. Secondly, the method is also shown to work at higher temper- atures (≈70 ◦C) than previously demonstrated (cryogenic and room temperatures). As the experimental environments are not as strictly controlled here, some discussion is also had about how to increase the quality of the measurements. In addition, the temperature sensing is shown to work simultaneously together with a distributed vibration sensing technique developed earlier at RISE, using the same φ-OTDR setup and test fiber.
Chapter 2
The phase-sensitive optical time domain reflectometer
In this chapter, the working principles of a φ-OTDR as well as how to use it to measure quantities such as temperature and strain, are explained. To do this, an overview of how regular OTDRs work is given at first.
2.1 Rayleigh scattering in optical fibers
Rayleigh scattering occurs due to inhomogeneities that are smaller than the laser wave- length in the fiber material. These ubiquitous fluctuations in density or glass composition end up in the fiber during fabrication in a disordered and random fashion, and are re- ferred to as scattering centers. As part of the incident electromagnetic waves interact with these scattering centers, the centers act as dipoles that start oscillating with the same frequency as the incoming light. Since oscillating dipoles radiate, this gives rise to light emitted in random directions with the same frequency as the incoming light, i.e.
elastic scattering. Because the fiber only guides light going in very specific directions (along the fiber axis), most of this scattered light is lost as it is sent outwards from the core into the cladding. This means that the light passing through the fiber is constantly being attenuated, decreasing exponentially in power P as it goes, according to
P (z) = P0e−αz, (2.1)
where z is the spatial coordinate along the fiber and α is the strength of the losses.
While there are other sources of losses in modern single mode fibers, the vast majority does come from Rayleigh scattering at the wavelengths we are interested in. As the intensity of Rayleigh scattering depends on the optical wavelength λ, the portion of the losses stemming from this phenomenon αR can be found as [5]
αR = C
λ4, (2.2)
where C is a material dependent constant with values typically around 0.7–0.9 (dB/km)µm4 for optical fibers. The wavelength of the light used is thus of importance to this technique.
However, a small percentage of the scattered light is re-emitted in a direction that the fiber supports. Half of this will be in the original direction of the incoming light,
effectively making no change to it, and half of it will be in the opposite, backwards direction. This is referred to as backscattering, and it is the key phenomenon in optical time domain reflectometry. Since the scatter is uniformly distributed in all directions, the fraction η of the scattered light that is recaptured and sent back can be estimated by finding the fraction of directions in which the fiber guides backwardly. To do this, we note that the steepest angle to the fiber axis θ that guided light can have in a single mode fiber is given by
NA = n1sin θ, (2.3)
where NA is the numerical aperture of the fiber and n1 is the refractive index of its core. We then take the solid angle subtended by the cone with apex angle 2θ, given by Ω = 2π(1 − cos θ), making the fraction we seek
η = Ω
4π = (1 − cos θ)
2 . (2.4)
Inserting typical values of NA (≈ 0.1) and n1 (≈ 1.445) for single mode fibers, this fraction is of the order of
η ≈ 10−3. (2.5)
With a typical Rayleigh scattering loss of 0.2 dB/km, corresponding to a fractional loss of 4.6×10−5 m−1, the backscatter reflection coefficient then amounts to −73.4 dB/m.
It is thus crucial to employ proper amplification and sensitive detectors to be able to detect this very weak signal. It is also important to minimize reflections that might occur in systems making use of this phenomenon, as they can easily drown out the signal or saturate detectors.
2.2 The principles of an OTDR
An OTDR works by sending out short optical pulses, typically on the scale of nanoseconds
— giving the pulses a spatial width on the scale of meters — into a fiber under test (FUT).
The device then measures the backscattered light coming back from the fiber, and plots the intensity against time. Since the speed of light in the fiber is known, it is then easy to link the reflected intensity in the time domain to its spatial position of origin along the fiber.
By doing so, it is possible to infer information about the fiber’s characteristics. As light that is backscattered further down along the fiber has to travel for longer through the lossy medium, it will have a weaker intensity when it reaches the detector. In a fiber without defects for example, the optical attenuation per unit length will be constant along the fiber. This would show up as a straight line with a constant slope (proportional to the losses) when plotted logarithmically. See Fig. 2.1 for a typical trace.
Features on the fiber such as bends, splices, connectors, defects, and the fiber end will also show up in various ways. For example, bends and bad splices will introduce additional localized losses into the fiber, which give rise to sudden drops in the intensity at their respective positions. Likewise, the glass-air interface at the end of the fiber will cause relatively strong reflections, usually around 4%, or −14 dB, of the exiting light.
The same is true for connectors, which lead to spikes of intensity at these positions in the plot — followed by drops as well, as reflections necessarily lead to losses.
Figure 2.1: Typical OTDR trace, with some common features. The attenuation in the fiber can be deduced from the slope of the graph.
The pulses are usually sent out and detected at a high rate, updating the plot in real time and giving the user a practically continuous view of the fiber characteristics where subjecting the fiber to a disturbance such as a bend would immediately show up as a drop. There is however a limit to how high this refresh rate can be. Since the speed of light in a fiber with refractive index n and length L is given by c/n, the time it takes for a pulse to be emitted from the source, be reflected at the end of the fiber, and finally come back to the detector can be found as T = 2nL/c. This means that the refresh rate f with which the device updates can be no higher than
f = 1 T = c
2nL, (2.6)
lest backscatter from different pulses starts mixing. For example, with a fiber of length 5 km the refresh rate should be kept below 20 kHz.
The smallest distance between two points that the device can distinguish is limited by the duration of the pulse. Suppose we are interested in knowing the width of spatial coordinates from which light that was detected at a given time could have originated from. The furthest point away must have been backscattered from the forefront of the pulse, while the closest point must have come from the end of the pulse. See Fig. 2.2 for a visualization. In order for the light at the back of the pulse to arrive synchronized with the front light, it must backscatter a small time nd/2c later, where d is the spatial width of the pulse. Thus, the width we are looking for is given by d/2. Of course, the time resolution of the detector also matters. If the smallest time window that the detector can distinguish is given by ∆t, this adds a further spatial width of ∆z = 2nc ∆t.
OTDRs are typically used in industry to characterize long fibers, find faults and de- fects, or localize bend losses. Large part of their usefulness stems from the fact that access to only one of the fiber ends is required, as the fibers can often be several kilo- meters long in these scenarios. The effective range of the device is ultimately limited by
z t
a b
Pulse Backscatter
d/2 d
Figure 2.2: Spatial resolution of the OTDR. This graph shows the pulse of width d prop- agating through spacetime, causing backscatter at points a and b. The backscatter from these distinct points (separated spatially by ∆z = d/2) will arrive at the detector at the same time, making them indistinguishable to the system.
the dynamic range, i.e. the ratio between the strongest non-blinding intensity and the weakest detectable intensity. As the pulse travels along the fiber it gets more and more attenuated, and its backscatter weaker and weaker, until it disappears into the noise floor
— usually the pulse reaches the end of the fiber before this happens though, barring any major losses on the way.
2.3 The principles and measurement capabilities of φ-OTDRs
The only real difference between an OTDR and a φ-OTDR is the linewidth of the opti- cal source. Whereas an OTDR uses a broadband light source, the φ-OTDR specifically makes use of a high coherence in the pulses and therefore needs a narrow linewidth.
The consequence of this improved coherence is that the backscattered light is allowed to constructively and destructively interfere with itself, giving rise to fringes in the detector trace. The characteristic straight line of the OTDR is thus replaced by an interferomet- ric pattern with an abundance of local minima and maxima, see Fig. 2.3 for a typical appearance.
Whether a given spot will interfere constructively or destructively is determined by the configuration and abundance of nearby scattering centers (contributing backscatter with differing phase, hence the name) which are, as mentioned earlier, distributed in an unpredictable and disordered manner. The exact shape of this ’interference pattern’
Figure 2.3: Typical appearance of a φ-OTDR trace. On average, the intensity falls off exponentially just as in the regular OTDR case. On the length scales used here this slope is not visible though, especially considering the fringes.
consequently follows no distinct pattern at all and will vary between different positions and different fibers.
2.3.1 ∆n and restorability
To formalize the above a little bit, suppose we want to know the difference in phase
∆φ between backscatter originating from two different scattering centers separated by a small distance ∆z. Remembering that the light has to cover the distance twice, this can easily be found as ∆φ = 2∆zωn/c, or equivalently,
∆φ = 22πc λ ∆zn
c ∝ n
λ, (2.7)
where ω is the angular frequency of the laser and λ its wavelength. What is interesting here is not ∆φ in itself. After all, making use of it would require information about all nearby contributing scattering centers which is simply not feasible. Instead, what is interesting is the fact that the expression is proportional to the factor n/λ. This means that the detected intensity I not only changes as you move along the fiber, it also changes with the laser wavelength as well as the refractive index, I = I(z, λ, n), see Fig. 2.4 for a comparison between two different wavelengths. Most importantly however, it means that if n is suddenly changed by a small amount ∆n (say, by an increase in temperature), a corresponding shift in laser wavelength
∆λ = λ
n∆n (2.8)
will recover the previous phase differences, restoring the earlier interference, and with it, bring back the same backscatter intensity, I(z, λ, n) = I(z, λ + ∆λ, n + ∆n). This is a very important feature referred to as restorability. As we shall see, it is what the temperature sensing technique relies on to work.
Figure 2.4: Comparison between two nearby wavelengths at the same fiber position.
Even though the wavelength is only 0.8 pm higher in the red trace than in the black trace (λ0≈ 1533 nm), the same position along the fiber shows a quite different shape.
2.4 How to measure ∆n
In principle, it is possible to measure ∆n by saving a reference trace I0(z) = I(z, λ0, n0), allowing n to shift, and then sweeping the wavelength until a matching trace I1(z) = I(z, λ1, n1) = I0(z) is found. ∆n is then easily retrieved by inverting equation 2.8. There are some problems with this approach, however. Firstly, it assumes that the shift in the refractive index is uniform over the fiber or its section of interest, which is rarely the case. Generally ∆n = ∆n(z), making I = I(z, λ, n(z)) a more practical way of seeing the intensity.1 In fact, n is constantly undergoing small, but in this context easily noticeable, and local fluctuations, largely due to acoustic vibrations from the surroundings and subtle shifts in ambient temperature, which makes this approach even less viable. These fluctuations tend to exhibit both high frequency behavior, comparable to or higher than the rate at which the probing pulse is sent, as well as lower frequency behavior on the scale of seconds or even minutes, i.e. drifting. This is discussed further in section 3.6.
This simplistic method also requires the laser to be able to make very small steps in wavelength. A more sophisticated approach is thus needed.
2.4.1 The correlation method
In order to measure the shift in the refractive index, it is helpful to move away from thinking of the intensity as something that varies in the z-direction as it is presented in the previous figures and instead think of it as something that, for a given z-coordinate and refractive index, varies in the λ-direction, i.e. I0(λ) = I(z0, λ, n(z0)). As we have seen earlier in Fig. 2.3, I follows no particular pattern in the z-direction. The same is true in the λ-direction (or equivalently, when varying the optical frequency ν), see Fig. 2.5 for
1The point is to make a distributed measurement, after all.
Figure 2.5: Appearance of the intensity function at a given point z when varying the optical frequency ν. Note that the resolution here (100 MHz) is a bit too poor to make out the proper features of the function, which exist on a scale slightly smaller [3]. As will be shown however, this resolution is mostly sufficient for the purposes of this project.
an example.
What is useful about this perspective however, is that, in accordance with equa- tion 2.8, a shift in refractive index ∆n simply appears as a lateral translation of the function, I(λ) → I(λ − ∆λ), instead of a complete change of shape as it did in the z-domain (see Fig. 2.4). Actually, presenting the change in I(λ) as a purely lateral shift is slightly misleading, as a difference in n will actually scale the λ-axis rather than trans- late it, see equation 2.7. However, for the small differences in n and λ that appear in this project (relative changes on the scale of ∼ 10−5), the first order approximation, i.e.
seeing it as lateral shift, is a completely satisfactory approximation and a useful point of view that helps the intuitive understanding of the method. This is a direct consequence of the restorability just explained.
The idea here is to, for a given z, map part of the I(λ)-function both before and after allowing n to change, and then cross-correlate the two in the λ-domain. This will give a correlation function C(∆λ) that, under good conditions, will display a distinct peak at the ∆λ that corresponds to the change in n at that z, see Fig. 2.6 for an example using artificially generated data. One can then simply retrieve ∆n by determining the location of the peak.
In practice, this mapping of the I-function is done by scanning the laser wavelength in small steps and recording the oscilloscope traces one by one. This means that the process is done in parallel along the whole fiber, and the result is a two-dimensional correlation function that depends both on ∆λ and z, C = C(z, ∆λ). The correlation peak’s position along the ∆λ-axis depends on ∆n, which as argued depends on z. This gives C a surface that resembles a mountain ridge, with a continuous peak of somewhat constant height along some line or curve in the (z, ∆λ)-plane. ∆n(z) can then be found by following this ridge and inverting equation 2.8 for each z. In this project, heat maps are used
Figure 2.6: Left: Visualization of shift in λ-domain induced by shift in n. Note that the red trace is the same as the black trace, but shifted a distance ∆λ∗ to the right. The data in this graph has been generated to emphasize the concept, as it is hard to make out in the real data due to its low resolution.
Right: Cross-correlation between the two traces. The correct index shift can be retrieved from this by identifying the peak at ∆λ∗ and using equation 2.8.
rather than actual 3-dimensional plots as they are easier to overview. See Fig. 2.7 for a conceptual sketch. Note that a higher shift at some z does not necessarily mean that the refractive index is higher than in the neighboring regions — only that it was higher in the second mapping than the first one at that spot. This means that one cannot make inferences about differences in n between different sections of the fiber unless it is known that the index was the same or had some known non-zero difference during the first mapping.
z 𝛥𝜆
Low High
0
Correlation Heat Map
Figure 2.7: Sketch of an example correlation heat map. The cyan streak of high cor- relation should follow the relation ∆λ(z) = λn∆n(z). In other words, horizontal lines of high correlation at ∆λ = 0 correspond to sections of the fiber where the refractive index has not shifted between mappings, whereas the correlation will be high some distance away from the z-axis where there has been a shift.
2.5 The temperature dependence of n
Since the system actually measures a difference in refractive index ∆n rather than a direct temperature change, a quantitative link between the two is needed. At room temperature, the sensitivity in typical fused silica can be found as [7]
dn
dT = 8.57 × 10−6 K−1 (2.9)
at λ = 1.5 µm. However, the experiments in this project are made at higher temperatures (60–70 ◦C), which is slightly outside the range of analyzed temperatures in [7]. Making a rough extrapolation with the data available puts the sensitivity closer to
dn
dT ≈ 9.8 × 10−6 K−1 (2.10)
at the temperatures and wavelengths in question.
Using equation 2.8, we can now relate a shift in temperature to the correct shift in wavelength. With λ = 1532 nm and n = 1.445, we expect to see a sensitivity of about2
dλ dT = dλ
dn · dn dT = λ
n dn
dT ≈ 10.4 pm/K. (2.11)
Note that this is close to the sensitivity typically seen in fiber Bragg gratings (FBGs) [8], as they basically work through the same interplay between wavelength and refractive index.
2.6 The strain dependence of n
The refractive index in the fiber does not only depend on its temperature, it is also affected by mechanical stress factors such as strain, defined as
e = ∆L
L , (2.12)
where L is the original length of a fiber section and ∆L is its increase in length under stress. Here we use the unit microstrain, denoted by µε, to refer to a strain of 10−6, i.e.
1 µε ⇔ e = 10−6.
In principle, this means it is not possible to know whether or not it is temperature being measured or something else, which could be problematic in certain scenarios. Some efforts have been made to get around this, for example through the use of birefringence [3].
Most of the time it is obvious from context what is being measured though, and for the scope of this project it is sufficient to test the strain measurement capabilities and attempt to find a plausible sensitivity. Here too we expect the relationship to be close to that of an FBG, which is typically around [8]
1 n
dn
de = 0.78 × 10−6 µε−1, (2.13)
2This does not take into account the change in optical path length between scattering centers due to thermal expansion. That effect is quite negligible in the context of this rough estimate, though.
making the expected wavelength shift relation dλ
de ≈ 1.2 pm/µε. (2.14)
Making general statements about strain sensitivity is however more complicated than its temperature counterpart, as factors like fiber coating play a big role. For this reason the numbers here might not correspond as well to the measured amounts in later sections.
Chapter 3
Experimental validation
In this chapter, the concepts explained in the previous chapter are applied and explored experimentally. First, the setup of the φ-OTDR is explained, and a measurement protocol for its use is presented. Some insight into how the gathered data is handled is also given. Then, four measurements are shown and their results are briefly discussed. The first one characterizes the temperature sensitivity of the system, while the second one performs a temperature measurement in improved environmental conditions. The third measurement attempts to characterize the strain sensitivity, while the fourth one tests the compatibility with simultaneous distributed vibration sensing in the same fiber. Lastly, some discussion is had on how to improve the results.
3.1 The setup
To validate the temperature measuring capabilities and give a proof of concept, a φ- OTDR was constructed. See Fig. 3.1 for a schematic of the setup.
Laser SOA
EDFA 1
EDFA 2
#1 #2
#3
FUT (SMF) Computer
Oscilloscope
Optical Electric Driver
APD VA Filter WG
Figure 3.1: Schematic of the φ-OTDR. SOA: semiconductor optical amplifier, EDFA:
erbium-doped fiber amplifier, FUT: fiber under test, VA: variable attenuator, APD: avalanche photodiode, WG: waveform generator. The details of the FUT can be found in the following sections.
The setup consists of a number of components. Following the path in the figure, we first have the tunable laser and its driver that delivers CW light. The wavelength
is controlled by a computer that also stores the data. Immediately after the laser, an optical isolator is placed to avoid optical feedback in the laser cavity.
Next up, the beam enters the semiconductor optical amplifier (SOA) driven by a pulse generator with a refresh rate of 20 kHz, which amplifies and shapes the beam into distinct pulses with a duration of around 10 ns, corresponding to a spatial resolution of around one meter. With a fiber length of less than 5 km, this refresh rate ensures that there is never more than one pulse in the system at any given time, see equation 2.6.
Then the pulse is amplified by an erbium-doped fiber amplifier (EDFA) to increase the input signal to the FUT, as the Rayleigh backscattering is expected to be weak.
Following this, the pulse then enters port #1 of an optical circulator, and is launched into the FUT via port #2. In this project a standard single mode telecom fiber is used. As the pulse makes its way towards the end of the FUT where it is terminated, it backscatters continuously, creating the longer, drawn out signal that self-interferes on its way back to port #2 on the circulator.
The Rayleigh backscatter is then collected from port #3 after which it is amplified by another EDFA, as the signal is still comparatively faint. Next, it passes through a tunable optical bandpass filter. This component is needed because EDFAs generate a lot of amplified spontaneous emission (ASE) noise. This noise is incoherent in nature and exists over a broader spectrum, and was it not filtered out it would drown out the signal.
The 1 nm wide filter is thus tuned to the laser wavelength, removing the majority of the ASE noise.
Then, the signal goes into a variable attenuator to avoid saturating the subsequent detector. An avalanche photodiode (APD) was used because of its speed and ability to detect weak signals. The photo detector converts the intensity of the light into voltage and feeds its signal to an oscilloscope that displays and updates the trace in real time.
The signal from the oscilloscope is acquired using a Labview virtual instrument (VI), which stores the data and prompts the laser to change its wavelength at the correct intervals. The oscilloscope is also connected to the waveform generator which acts as its trigger, making sure that it consistently refreshes at the same rate that the backscattered pulses arrive at. In addition, it uses an adjustable moving average that displays the average shape of a number of the latest traces. This average is made before the data is sent to the computer, which reduces the amount of data that needs to be transferred and saves time in the processing, but inhibits any statistical analysis of the individual traces.
3.1.1 The measurement protocol
As explained in section 2.4.1, two mappings of the intensity function I(z, λ, n(z)) need to be done, before and after allowing n(z) to change, in order to compare them and to find
∆n(z). One can of course allow n(z) to shift further and make more maps, allowing for comparisons between any two maps in the complete set. This is how the measurements in this project are done in general, as it is an easy way to increase the amount of data.
Note that the oscilloscope must be kept to the same section and amount of zoom for all mappings in the same batch for this to work.
To complete one mapping, the specific environment of the FUT is first prepared, and the oscilloscope is configured to display the section of interest, with its time axis set to show 600 ns. Regardless of the degree of zoom, the device always displays 1200 data points along the time axis (essentially the z-axis), and it is these points that are sent to
the computer. This means that one has to make a choice of which section of the FUT is of interest before starting the measurement with this particular setup (zooming out too much to include the entire FUT risks losing out on detail of the trace, although the effects on the correlation of doing this was not explored quantitatively). See Fig. 3.2 for a flowchart of the procedure.
λ λ
Figure 3.2: Flowchart of the measurement protocol. The first two steps are repeated a few times before plots are made.
Then, the software on the computer that controls the laser and oscilloscope is started.
It sets the tunable laser to its starting wavelength, corresponding to an optical frequency of 195.660 THz, and then waits 15 seconds to allow ample time for the laser to stabilize.
After waiting, the program reads the current trace shown in the oscilloscope, which is an average of the last 128 individual traces to enter the device,1 and saves it to a file. Then, the program increases the optical frequency by 100 MHz and again waits for the laser to stabilize, and repeats this procedure. In the end, 301 different frequencies are recorded for one map, taking about 75 minutes to complete.
To then make more maps in the same batch, one simply alters the environment of the FUT and starts the software again. For a more complete collection of the measurement parameters, see table 3.1.
3.1.2 Post processing
When the mappings are completed, they are run in pairs through a MATLAB script whose main role is to compute the correlation function C(z, ∆λ) between them. First however, the mean is removed from each slice in the λ-direction of the intensity function maps, i.e.
I(zi, λj) → I(zi, λj) − 1 N
N
X
k=1
I(zi, λk), (3.1)
where N is the number of wavelengths in the map. If this is not done, the (comparatively) constant noise floor of each trace will later contribute a lot of positive correlation to C(z, ∆λ) that we are not interested in.
1As the device refreshes with a rate of 20 kHz, this means that the averaging takes place over a time span of 6.4 ms.
Table 3.1: Parameters of the φ-OTDR and the measurements made with it.
Property Value Corresponding Value
Starting frequency 195.660 THz ∼ 1533 nm Frequency step size 100 MHz ∼ −0.8 pm Number of frequencies 301
Total scan range 30 GHz ∼ 240 pm
Wait time between steps 15 s
Mapping time 75 min 15 s
Averaging amount 128
Refresh rate f 20 kHz
Time span of averaging 6.4 ms
Oscilloscope time span 600 ns 62.1 m Data points along time axis 1200
Pulse length / Resolution ∼10 ns ∼1 m
Next, the discrete cross-correlation function C(zi, ∆λj) is computed. Giving the dif- ferent mappings subscripts 1 and 2, it is calculated as
C(zi, ∆λj) = 1 A(zi)
∞
X
k=−∞
I1(zi, λk)I2(zi, λk+j), (3.2)
where I is taken as 0 where the wavelength index goes out of bounds, and A is a nor- malization factor given by
A(zi) = v u u t
N
X
k=1
I1(zi, λk)2
N
X
k=1
I2(zi, λk)2. (3.3)
Note that the expression is normalized such that a perfect correlation at zero shift will return a value of 1, even if the traces are weaker during one of the mappings. Also note that the index j of the shift ∆λ is allowed to take on values from −N + 1 to N − 1 (rather than values from 1 to N as for the absolute wavelength in the maps), as both positive and negative shifts are possible.
Something to be careful with here is the function’s behavior for large shifts, i.e. as
|j| → N − 1. Here, the function will always return correlations close to zero, even if it is the correct shift, due to the decreasing non-zero overlap between I1and I2 in equation 3.2.
It also increases the probability of high correlation at an erroneous shift (as in comparable to or higher in strength than a correct shift with |j| ≈ N − 1) since the amount of points that have to randomly match are lower. Something which, due to the wavy nature of the intensity function, is quite likely to happen at a few places. This is why it is important to scan wavelength intervals much wider than the expected measured shift — we want the continuous peak of the function to be relatively close to the z-axis so that it properly rises above the floor of ’correlation noise’.
Finally, the correlation function is plotted as a heat map in the (z, ∆λ)-plane such as the one in Fig. 2.7.
𝝓-OTDR
Spool 1 Spool 2 Spool 3
~20 m Oven
Oscilloscope view
Figure 3.3: Schematic of the FUT environment used in the temperature sensitivity characterization. Spool 1 is placed in normal room temperature conditions, Spool 2 holds 18 meters of fiber and is placed inside an oven together with a thermocouple. The oscilloscope view is set to survey 60 meters of fiber, starting at the last ∼20 meters of Spool 1 and ending a few meters after Spool 2.
3.2 Characterization of temperature sensitivity
FUT environment
The first experiment served as a calibration of the temperature sensitivity of the fiber, noting down the real temperature differences and looking at the how big the shifts were.
The FUT consisted of a few different parts: a spool placed on the bench in the lab as reference (where we expect little or no shift), fiber running down a ∼20 m corridor, and a spool placed inside an oven with varying temperature. See Fig. 3.3 for a schematic.
After the oven, an additional spool was placed to create some distance from the fiber end in case of any reflections. To reduce reflection effects further, sharp bend losses were introduced after the spool, and the very end of the fiber was placed in a vial of index matching liquid. Since part of the FUT had to be able to handle higher temperatures, the section after the first spool consisted of polyimide-coated fiber as opposed to the beginning of the fiber which had the typical acrylate coating.
As the oven was designed to create temperatures in the high hundreds of degrees Celsius, it was not very precise in the temperature range tested here, around 60 ◦C.
For this reason, a thermocouple with a resolution of 0.1 ◦C was placed inside the oven with the fiber, and the actual temperature T was noted down in conjunction with the mappings. For the temperatures in question, see table 3.2. The spool placed on the bench was simply allowed to sit in ambient room temperature, with all the fluctuations and drift that it entailed.
From all this, we expect to see correlation functions with high values at ∆λ = 0 for the bench spool and corridor part, and strong correlation in a straight line for some non- zero value of ∆λ in the oven part. The heatmaps should mostly look like Fig. 2.7 from before, consisting of horizontal lines of high correlation with different distances from the z-axis.
Table 3.2: Oven temperatures T measured with the thermocouple during mappings in sec- tion 3.2.
Map Number T
(◦C) 1 62.1 (±0.1) 2 65.8 (±0.1) 3 67.4 (±0.1) 4 68.2 (±0.1)
Results
The mappings were ran through the MATLAB script in pairs and the correlation func- tions were plotted, see Fig. 3.4 for an example. In the figure we see a streak of heightened correlation at the position of Spool 2 inside the oven, towards the end of the view. Here the refractive index has increased between the mappings as the shift in the ∆λ-direction is positive. The other sections display no significant correlation compared to the noise however, which was typical for the correlation functions in this section. As to not in- troduce any bias towards any one particular mapping, all possible combinations were evaluated this way. From this, the shifts in the hotspots were visually extracted and saved, see table 3.3.
Figure 3.4: Correlation function between Mappings 2 & 3 from section 3.2. The graph has been zoomed in in the ∆λ-direction (the full domain limits are ±240 pm), and the color scale has been manually set to increase contrast, where values outside the limits are displayed with the limit colors.
With this data, the temperature sensitivity of the device was estimated through linear regression to be 11.18 (±0.89) pm/K, see Fig. 3.5 for the fit. This is slightly higher than expected, but the uncertainties do include the expected value of around 10.4 pm/K.
Table 3.3: Hotspot wavelength shifts ∆λ between mappings in the temperature sensitivity characterization, with associated temperature differences ∆T .
Compared Mappings ∆T ∆λ
(◦C) (pm) 1 → 2 3.7 (±0.2) 43 (±1) 1 → 3 5.3 (±0.2) 58 (±1) 1 → 4 6.1 (±0.2) 68 (±1) 2 → 3 1.6 (±0.2) 14 (±1) 2 → 4 2.4 (±0.2) 25 (±1) 3 → 4 0.8 (±0.2) 11 (±1)
Figure 3.5: Linear fit of temperature sensitivity, using data from section 3.2.
Discussion
While there are clearly streaks of higher correlation at the correct spots in the oven parts of the results, it is still quite weak and not significantly stronger than the surrounding noise. At a few points in the correlation function, the amplitude of the noise even surpasses that of the correct shift. In this particular test it is not a huge problem since the human eye is good at recognizing the simple pattern of a straight line (i.e. constant shift along some fiber section), and it is as such easy to see where the correct shift is supposed to be situated, but for a more general temperature distribution this lacking contrast between noise and signal would make it unfeasible to find the real temperature change at every point along the fiber.
Furthermore, the sections of the fiber situated in room temperature outside the oven showed little to no heightened correlation at all. This is obviously a huge problem since it completely kills the method in these sections. Comparing the two environments, there are two main differences between them that could explain this variation in performance.
The first is the absolute temperature. This explanation seems unlikely though, as the method has already been shown to work in room temperature [3]. This leaves the second difference, which is fiber isolation. Firstly, the temperature inside the oven is bound to
be more consistent and deviate less as it is quite a controlled environment in this regard.
The temperature of the fiber on the bench probably experiences bigger fluctuations — while the lab is climate controlled, the rooms are big and windows provide opportunity for drifts in temperature due to uneven heating or cooling in the hour or so needed for one mapping. As shown in [3], this method is very sensitive and can detect temperature changes on the mK scale, which explains why these comparatively small fluctuations could have an effect. Secondly, the fiber outside the oven is a lot more exposed to acoustic vibrations. It is well documented that the φ-OTDR technique is fully capable of picking up these types of vibrations [1, 4], so it is no surprise that this might have an effect on measurement quality. In the oven, outside vibrations are attenuated as they pass through its walls and the only real source of noise is the consistent hum of the oven itself. The environment around Spool 1 is less controlled since other experiments are being performed simultaneously and the fiber is less isolated from external perturbations.
More discussion on how to remedy this is had in section 3.6.
With all this noted, it was still possible to extract the temperature sensitivity from the oven spool, and it was found to be quite close to the expected value.
3.3 Measuring temperature
FUT environment
The second experiment sought to make an actual temperature measurement, using the sensitivity obtained during the temperature characterization together with a measured shift to estimate a real temperature difference. The setup was largely the same as in the previous characterization with one main difference, see Fig. 3.6. This time, an additional spool was placed in a second oven, and the first oven (now labeled Spool 1) was kept at a constant temperature of 70◦C to be used as the reference instead of the room temperature spool. This was done because of lacking performance in the room temperature spool in the previous measurement (see also section 3.6). Just as before, the temperatures were also measured by the thermocouple to see how well the fiber measurement did, see table 3.4.
𝝓-OTDR
Spool 2 Spool 3
Oven
Oscilloscope view Spool 1
Oven
Figure 3.6: Schematic of the FUT environment in section 3.3. The first spool in the previous section has been replaced with a spool inside a second oven. Spool 1 is kept at constant temperature while Spool 2 is subjected to different temperatures. The oscilloscope view is still set to 60 meters, surveying Spool 1 & 2, as well as some distance before and after them.
Table 3.4: Oven temperatures T measured with the thermocouple at Spool 2 during mappings in section 3.3.
Map Number T
(◦C) 1 71.5 (±0.1) 2 74.3 (±0.1)
Results
The wavelength shift between mappings 1 and 2 were measured to be 29 (±1) pm, see Fig. 3.7. Using the temperature sensitivity obtained in section 3.2, this gives a temper- ature difference of 2.6 (±0.3) K measured by the fiber, as compared to a temperature difference of 2.8 (±0.2) K as measured by the thermocouple (from table 3.4).
Additionally, the correlation function between these mappings is a lot better than the previous ones. The graph is now much closer to the expected result from Fig. 2.7. Here the contrast is much higher around Spool 2 and the correlation clearly rises above the noise. Although the correlation is weaker around the reference sections, there is clearly a line along ∆λ = 0 where it should be.
Figure 3.7: Correlation function between Mappings 1 & 2 in section 3.3. Although the upper limit of the color scale is set to 0.25 for contrast reasons, the correlation function assumed values over 0.5 at the hotspot.
Discussion
The system managed to pick out a temperature within the margin of error, validating the procedure. What might be more interesting to note however, is that the quality of the correlation function was a lot better this time than in the characterization. This goes
for both at the hotspot where the temperature was increased — here we see a clearer, tighter peak — as well as the other section where it stayed more constant, where we before did not see heightened correlation at all. It is possible that the second oven (the new hotspot) is better at holding the temperature constant than the previous one, or it is simply less noisy. As for the streak at ∆λ = 0, this improved performance can be explained by the fact that it is now situated inside the first oven. In general, there also seems to be a variance in performance between measurements done under similar conditions. Part of this is because the environment is not fully controlled, but it could also have to do with a lacking frequency resolution in the scanning laser, more on this in section 3.6.
3.4 Characterization of strain sensitivity
FUT environment
The third and last measurement aimed at finding the strain sensitivity of the fiber. Here, the ovens were replaced by an Optiphase PZ2-SM2 Fiber Stretcher connected to a DC voltage source, see Fig. 3.8 for a schematic. The stretcher contains 40 meters of single mode fiber, and extends the fiber 3.8 µm per volt applied to the device, giving a strain of roughly 0.1 µε/V [9]. Because a quick preliminary test showed weak correlation, the amount of averaging was increased from 128 to 1024 for this experiment. The view in the oscilloscope was also zoomed in further, now displaying a time span of 240 ns, corresponding to 24.8 m of fiber, to see if this increased performance. For the voltages applied, see table 3.5.
𝝓
Figure 3.8: Schematic of the FUT environment in section 3.4. The whole setup is placed at normal room temperature conditions.
Table 3.5: Voltages applied to the stretcher during mappings in section 3.4.
Map Number Voltage (V)
1 0 (±1)
2 60 (±1)
3 80 (±1)
4 100 (±1)
5 150 (±1)
6 250 (±1)
Results
Here, all possible comparisons of the mappings were made just as in the previous sections.
This time however, the correlation was either very weak or not visible at all in many of the comparisons. See Fig. 3.9 for some examples. The shifts that were seen can be found in table 3.6.
Figure 3.9: Two correlation functions from section 3.4 (Left: 4 → 6, Right: 5 → 6).
No significant streaks of correlation can be seen in the left figure, which was the case for the majority of comparisons in this characterization. In the right figure, a streak can be seen, but it is weak in strength and appears to be smudged.
Table 3.6: Hotspot wavelength shifts ∆λ between mappings in section 3.4, with associated voltage differences ∆U and strains. Comparisons that gave no visible streaks of correlation are omitted.
Compared Mappings ∆U Strain ∆λ
(V) (µε) (pm)
1 → 2 60 (±2) 5.7 (±0.2) 6 (±2) 2 → 3 20 (±2) 1.9 (±0.2) 4 (±2) 5 → 6 100 (±2) 9.5 (±0.2) 26 (±2)
From this, linear regression was made, see Fig. 3.10. The sensitivity was estimated to be 2.2 (±0.6) pm/µε, higher than the expected value of around 1.2 pm/µε.
Figure 3.10: Linear fit of strain sensitivity, using data from section 3.4.
Discussion
With only 3 out of 15 possible map comparisons showing any kind of correlation (and weak correlation at that), one would be hard-pressed to call this a successful calibration.
Furthermore, the obtained sensitivity is almost two times as large as predicted in sec- tion 2.6. With that said, deviations from that value were expected. Fiber coating plays a big role in the strain sensitivity, and the exact coating of the fiber inside the stretcher was not known, so it is hard to make statements for or against it being a contributing factor to the seen difference. In addition, the stretcher makes use of a coiled fiber, which might cause local differences in strain between windings. With a spatial resolution of 1 m and a coil diameter of ∼7 cm, these would get smudged together and blur the correlation peak. Something else to consider is the fact that the wavelength shifts we are looking at here are smaller than the shifts in the temperature sections, so the drift in ambient temperature mentioned earlier have a bigger relative effect on the measurement.
3.5 Simultaneous temperature & vibration sensing
An example where this distributed temperature measurement could be potentially applied in the framework of an industrial project, is for conveyor belt monitoring in mining industry. Another phenomenon that might be interesting to monitor in these types of situations is vibration. Through the analysis of the conveyor belt elements vibration frequency and temperature profile, one can detect and localize, amongst other hazardous phenomena, early stage bearing defects, belt misalignment and delamination, overloaded bearing, resonance events and overheating of belt components.
As mentioned earlier, this is an area that φ-OTDR technique has already seen use in.
RISE has already developed an in-house interrogation system for localizing and measuring vibration in a distributed way along kilometers of length of fiber [4]. This interrogator
has been tested in an industrial setting to measure the acoustic profile (different vibration amplitude and frequency) on a bearing when the latter was subjected to different loads and rotation speeds during operational conditions, and to detect sounds from idlers and pulleys from a conveyor belt. Additionally, the φ-OTDR setup used for temperature sensing here can also be used for this vibration sensing technique. This means that, with some modifications to the computer software, the system can measure temperature differences and acoustic profiles simultaneously in the same fiber. To give a proof of concept, an experiment was performed.2
FUT environment
The fiber stretcher was again used, this time driven by AC voltage at 60 Hz to introduce vibrations to the fiber inside it. As the temperature sensing had proven easier to do inside the ovens, a spool was placed inside one of them. See Fig. 3.11 for a schematic.
Note that the two measurement techniques focus on two different spots of the FUT, separated by ∼20 m. This is partly because the oven, whose temperature control was needed to create a change ∆n, is not a good place for the fiber stretcher, but also because the induced vibrations might weaken the correlation, as we have seen fluctuations do.
However, if the correlation is strong enough there is in principle no real reason that one cannot simultaneously measure both things at the same spot as well. More on ways to improve the correlation in section 3.6.
𝝓-OTDR
Spool 1 Spool 2
Oven
Oscilloscope view Fiber Stretcher
Figure 3.11: Schematic of the FUT environment in section 3.5. This time the fiber stretcher was driven with AC voltage at 60 Hz to test vibration detection.
Results
The intensity function was mapped at 42.1 ◦C and 45.3 ◦C (∆T = 2.2 K), and the acoustic profile of the fiber inside the stretcher was measured. See Fig. 3.12 for the results.
2For details on measuring vibrations and retrieving acoustic profiles, see [4]. Only the measurement results are presented here.
Figure 3.12: Left: Acoustic profile measured inside the fiber stretcher. A clear peak can be seen at the expected frequency, 60 Hz.
Right: Correlation function around the oven spool. A streak can be seen in the middle of the figure, showing the shift in n. The correlation is somewhat weak though, comparable to the measurements in section 3.2.
Discussion
While this measurement does not add new things about the temperature sensing capabil- ities in particular, it provides a proof of concept that the technique can be simultaneously compatible with other φ-OTDR applications without any hardware changes to the setup
— a property which might be quite useful in some cases.
3.6 Making better correlation measurements
There are of course a handful of factors that contribute to the data being less than optimal, weakening the correlation where it should be high.
One of these factors is the constant refractive index fluctuations δn caused by small ubiquitous changes in the environment surrounding the fiber, primarily acoustic vibra- tions [4] and small shifts in the ambient temperature. These will worsen the correlation since the intensity function will shift back and forth, making the mapping inaccurate.
Similarly, if the laser is imprecise (i.e. the step size varies a bit) or unstable and ex- hibits fluctuations in wavelength, this will have the same effect on the mappings. If these fluctuations are small, this can be solved by averaging traces just as described earlier.
However, in the case of big fluctuations averaging does not help much, even if the amount of averaged traces is drastically increased. This is because the intensity function is not monotonically increasing nor decreasing, and so averaging it over these larger intervals will simply yield the general average of the function rather than the value it assumes at the ’correct’ refractive index. See Fig. 3.13 for a visualization.
To prevent this, the fiber could be isolated from the environment, such that acoustic vibrations are attenuated and only temperature shifts that exist on longer timescales reach it. The smaller fluctuations that remain can then be handled by increased averag- ing. Here it might be worth thinking about the amount of time over which each trace is averaged. As noted earlier, this time is equal to 6.4 ms in our case. This implies that index fluctuations induced by vibrations slower than this (≈156 Hz, which in the acoustic case is well into the range audible to the human ear) are not averaged out. By instead
Figure 3.13: Comparison between small and large fluctuations δn. The green segments represent the width of the fluctuations. In the left figure, averaging the intensity over the green area will yield an accurate estimate of the value at δn = 0, whereas the doing it in in the right figure will simply yield something closer to the average of the entire intensity function. The data here has been generated to emphasize the concept.
averaging over one full second, which would be a small time compared the 15 s the laser currently waits each step, one should be able to get rid of all audible perturbations given they are not terribly strong in amplitude. For this particular setup, hardware limitations sets the maximum amount of averaged traces to 1024 per step, as opposed to the 20,000 needed for a full second. To get around this, one could lower the refresh rate of the waveform generator to 1024 Hz. This would affect the compatibility with the vibration sensing technique as it relies on a high refresh rate though, so averaging a higher amount of traces is preferable. If this is not sufficient one might have to restrict use to cases where these types of fluctuations are naturally weaker and less pervasive.
As mentioned in section 3.1, the traces are averaged in the oscilloscope before being sent to the computer. In a setup where all individual traces are available and this averaging takes place at a later stage, one could calculate the standard deviation of the measured intensity at any given point along the fiber. This could provide a way to evaluate if the fluctuations are too large, as a small standard deviation would indicate weak fluctuations. With the current system however, this is not possible and one has to attempt to evaluate things with more qualitative approaches.
Another way of improving the results is by using a laser capable of taking even smaller steps while sweeping the wavelength (the current one is capable of taking 0.8 pm steps at its smallest). When mapping the intensity function, all the features in the λ-direction should ideally be completely resolved. Otherwise, the true shift of the refractive index is not guaranteed to accurately correspond to an integer multiple of the wavelength step size, see Fig. 3.14 for an example. In the left graph of the figure, we can clearly make out the correct shift in the high resolution case, while it is far from obvious in the low resolution case. In the right graph of the figure, a distinct peak is visible at the correct shift, marked with a dashed vertical line, in the higher resolution correlation. For the lower resolution, the correlation at the correct shift is barely positive and it does not manage to rise above the noise — in fact, the highest peak is located at an entirely wrong shift. This is a case of the correct shift being right in between steps of large size as the correct shift was intentionally chosen here to be exactly half of the step size. For the higher resolution case, this is not be a problem, since its peak at the correct position
even has some width with many high values. The typical size of the features in the real intensity function is not obvious from the mappings done in this project, but looking back at Fig. 2.5 it is clear that the resolution is far from optimal.
Figure 3.14: Visualization of difference in performance between different step sizes.
Left: The ’true appearance’ of the intensity (generated data) can be seen in the middle, both before and after an index shift ∆n. Below, high resolution mappings with small wavelength step size of the respective intensities are shown with asterisks. Above, lower resolution samplings with steps of 7.5 times the size are shown with circles.
Right: Correlation function between mappings for the two different step sizes.
Another factor that inhibits strong correlation is mid-mapping drifting of n, i.e. if the external temperature changes substantially during one or both of the mappings. In the controlled environments of the lab ovens, this was not a big problem. With the long time a single mapping takes with this particular setup however, it is easy to imagine a scenario where the temperature moves in a way over time that completely prevents any correlation, or produces other problematic outcomes. Consider for example the case with a temperature T1 during the first half of mapping 1, followed by a temperature T2 = T1+ ∆T during the second half. Later, after half of mapping 2 is done, the temperature goes back to T1, i.e. it follows the reverse order. In this scenario, the correlation function would show peaks (of diminished height) at wavelength shifts corresponding to both +∆T and −∆T at the same positions along the fiber. A good way to remedy this would be to allow less time to drift by making quicker measurements, preferably by using a tunable laser that stabilizes a lot quicker after each step.
Another way to make the measurement faster is to take fewer steps in total. This is not a great approach however, as it worsens the measurement in two ways. Firstly, assuming the step size stays the same, it reduces the total scan range. This pushes the position of the correlation peak towards the edge of the function’s domain, which is bad for reasons outlined in section 3.1.2 (or rather, it pushes the edges of the domain towards the middle), effectively making the measurement range of the device smaller.
The alternative is to keep the same scan range by increasing the step size, but that is also bad as we have just seen. Secondly, even if the correct shift is still close to the middle, the lower amount of data will increase the noise floor, requiring a stronger correlation for it to stand out. See Fig. 3.15 for an example of correlation strength distribution with fewer steps. Note that, with the exception of a few outliers, using more steps pushes the entire distribution towards 0, and the histogram with 100 steps generally has a lot of positive correlation. However, the negative side, C < 0, provides a way to see the
general distribution of the noise (roughly outlined in the figure), which is a lot wider in the black graph. The useful correlation is that which appears significantly to the right of this outline. Consequently, having a higher correlation is not necessarily the end goal, the important part is getting contrast between the correct shift and the noise, which using many steps is a good way to do.
Figure 3.15: Histogram of values assumed by the correlation function, with different amounts of steps. The data is taken from section 3.3 (see Fig. 3.7), but only a portion of it is used in the black graph to simulate a mapping with fewer steps. Both histograms use the same input data and domain of the correlation function, meaning the total number of values (pixels) are the same.
Something else to consider here is how these proposed improvements interplay with each other. Decreasing the step size, for example, will decrease the total scan range, unless more steps are taken, in which case the measurement time is increased, which could prove problematic outside lab conditions. The same thing goes for averaging over longer times on each step, which might help filter out some of the slower fluctuations.
The optimal set of parameters is not obvious and probably varies between different use cases, but reducing the laser stabilization time is definitely a good thing and increasing the resolution from 0.8 pm is almost certainly bound to improve results.
