Dual-Wavelength Differential Detection of Fiber Bragg Grating Sensors: Towards a Sensor Ecosystem

Abstract

We discuss how the dual-wavelength differential detection (DWDD) of fiber Bragg grating sensors can be used to build standardized high-resolution, high-accuracy, large-measurement-range, multichannel instruments and associated sensors. We analyze the system resolution and experimentally show that the high signal-to-noise ratio can allow the design of sensors with a ratio of range to resolution superior to 14 bits, and temperature measurement ranges of more than 180 °C. We propose a scheme for real-time signal correction to cancel the drift of the instrument using two internal reference sensors, and a calibration method using centralized golden sensors that allows traceability to international standards for all instruments and sensors, allowing the creation of a global sensor/instrument ecosystem.

1. Introduction

Fiber Bragg grating sensors (FBGSs) have been the topic of innumerable studies over the past 35 years [1,2,3], as well as having found numerous commercial applications. FBGSs are made by inscribing a periodic refractive index pattern in a short section (typically 8–10 mm) of optical fiber using UV light. The FBGS reflects light over a narrow wavelength band that depends on the period of that pattern. Since the period is dependent on strain and temperature, a measurement of the spectral position of the reflected peaks readily provides a measurement of those physical quantities. The sensors are typically measured by analyzing the spectrum of the reflected light and detecting the peak locations. This is performed either with a broadband optical source and a spectrometer, or a swept-wavelength source and a photodetector.

Apart from the advantages intrinsic to all types of optical fiber sensors, such as immunity to electrical interference, FBGSs have been touted for their capability to be multiplexed [4], with multiple sensors written at different central wavelengths within a single strand of optical fiber, a technique called wavelength division multiplexing (WDM). In addition, a scheme called “time division multiplexing” (TDM) has been demonstrated [4,5,6,7,8,9,10], where the FBGSs are interrogated with short pulses, whose central wavelength is also swept. If the FBGSs are located at different distances, they can be distinguished by the arrival time of the reflected pulses. Finally, in addition to having multiple FBGSs in a single fiber, they can be located in separate fibers, with a separate photodetector used for each fiber. Such a scheme is called “space division multiplexing” (SDM) [11]. Combining those three multiplexing techniques allows systems with thousands of sensors to be built, interrogated with a single instrument [7].

However, when it comes to strain and temperature sensors, electrical sensors, mostly based on resistive or capacitive change, are, by far, much more widespread than fiber optic sensors in general, and FBGSs in particular. While cost is definitely an issue, it is only part of the problem. Fiber Bragg gratings (FBGs) can be and are actually mass produced at very low cost for other applications, such as pump diode stabilizers. But even if the cost of the instrument were to be reduced, FBGSs would not yet have the advantage of competing electrical-based sensors, which is a fair amount of standardization. For example, resistive strain gauges come in only a limited number of standardized models, and so do thermistors. They are already calibrated, either individually or from generic values (e.g., 10K thermistors) and can be used interchangeably with instruments that have themselves standardized characteristics. Load cells that use standard strain gauges are classified by the OIML (Organisation Internationale de Métrologie Légale) with precisely defined performance criteria. And while they cannot be multiplexed within single-wire strands like the FBGS, this is a property that is only useful in a limited number of cases. Thus, for the vast majority of temperature, strain, and other sensing applications, individual sensors use SDM, with instruments having from one to multiple ports. But to this day, such an ecosystem of standardized and calibrated instruments and sensors does not exist for FBGSs.

Yet, apart from multiplexing, FBGSs still have advantages over electrical sensors. One is the immunity to electrical interference. Resistive strain gauges, for example, are very sensitive to such interference, so long lead wires are not recommended, and their installation is tedious if one wants to avoid it. Resistive strain gauges also do not have very good long-term stability, while that of FBGS can be excellent. For temperature measurement, electrical interference is also an issue in applications such as transformer hot spot monitoring, but other fiber optic sensing technologies based on SDM, such as GaAs sensors, dominate that market instead of FBGSs. The main reason is that SDM is better suited for this due to the difficulty of placing sensors at different locations inside a transformer with a single fiber; FBGS instruments are not very well suited for SDM.

In order to leverage their intrinsic advantages and compete with electrical sensor technologies, FBGSs must therefore match their performance in terms of resolution and accuracy while being highly standardized so that both instruments and sensors can be used interchangeably with predictable calibration and performance.

Recently, a new technique called “Dual-Wavelength Differential Detection” has been proposed and demonstrated by various authors [8,10,12,13,14,15,16,17,18,19,20]. Although implemented in slightly different ways by different groups, the basic principle is to measure the FBGS at two slightly different wavelengths, within the FBGS bandwidth, and use the difference between the two measurements to extract the state of the sensor. The two wavelengths are fixed, and thus the system does not require a swept wavelength source or a spectrometer.

In particular, we have implemented DWDD by using the dynamic chirp of a single distributed feedback (DFB) laser diode to generate those two wavelengths at different points in time [14,15,16]. The dynamic chirp happens when the laser diode is either pulsed or modulated [21,22]. In such cases, the wavelength changes over the duration of the pulse or that of a modulation cycle. By digitizing the reflected signal, one can use samples at different times to perform the DWDD algorithm and extract the sensor status. We have demonstrated this both at high speed by using nanosecond pulses [16], in a TDM scheme, as well as by using a low-frequency (5 kHz) square wave modulation [15,17,18].

In our previous work [14,15,16,17], we found that DWDD can provide both a very high resolution as well as good accuracy and reproducibility. The instrument design in all cases is simple and low cost, consisting of a single standard DFB laser diode, with its driver, fiber couplers and photodetectors, as well as the ADC electronics. Though one scheme was made for TDM, it was used in an SDM configuration, where the sensors are located in different branches of an output splitter, with an increasing time delay added in each branch. In ref. [17], we introduced the use of a reference sensor to cancel the drift of the laser diode wavelength, which allowed us to obtain a spectral resolution of 40 fm, which is 50 times better than a typical optical spectrum analyzer (OSA).

DWDD is best suited to measure FBGS with identical design and central wavelength, which in itself opens the way to standardization. The high-resolution, long-term stability, and ability to use internal reference sensors for constant accuracy thus make it possible to build a standardized, high-performance sensor ecosystem. In this paper, we explore this possibility by presenting the initial performance results of an SDM-DWDD system, using low-frequency square-wave modulation, from which we estimate the best achievable performance with an optimum design for both sensors and instruments that can be realized with the available technology. We show how internal referencing can be implemented together with centralized calibration to provide constant accuracy, interoperability, and traceability to international standards.

Section 2 describes the basic principle of the DWDD method, and how the resolution can be calculated in the general case of a non-Gaussian FBG sensor profile, such as that of a uniform profile FBG. Section 3 reviews our previous implementations and describes our new SDM-DWDD system, presenting measurements of the dynamic chirp, stability and resolution. Section 4 shows how the design of the sensor can be made to maximize the range-to-resolution ratio by the proper choice of grating length, reflectivity, and instrument dynamic chirp. Based on our experimental estimation of the achievable signal-to-noise ratio (SNR), as well as achievable dynamic chirp, we find that instruments with more than 14 bits of resolution and a temperature measurement range of 180 °C can be designed. Section 5 introduces the concept of the twin-sensor internal reference cell and how it can compensate for the drift in the laser wavelength and dynamic chirp. It also proposes a procedure for the calibration of the sensors and the instruments that is extended to the use of a centralized “golden instrument” (GI) associated with two “golden sensors” (GS). We estimate the best performance that can be attained with this scheme, and discuss the requirement for the calibration accuracy and how those can be achieved with the current technology. Section 6 briefly discusses how the birefringence of the sensor is currently a limit to the achievable performance, and how this can be overcome.

2. DWDD: Basic Principle and Resolution

The basic scheme of DWDD studied in this paper, and used in our previous work, is illustrated in Figure 1. The reflected powers $P^{+}$ and $P^{-}$ are measured at two wavelengths ${\lambda }^{+}$ and ${\lambda }^{-}$ that have a separation typically smaller than the FBGS bandwidth. The sensing signal S is defined as:

$$S({\lambda }_s-{\lambda }_0)=\frac{P^{+}\left({\lambda }^{+}\right)-P^{-}\left({\lambda }^{-}\right)}{P^{+}\left({\lambda }^{+}\right)+P^{-}\left({\lambda }^{-}\right)}$$

(1)

where ${\lambda }_s$ is the FBGS central wavelength at the time of measurement, and ${\lambda }_0$ is the center wavelength between ${\lambda }^{+}$ and ${\lambda }^{-}$:

$${\lambda }_0=\frac{{\lambda }^{+}+{\lambda }^{-}}{2}$$

(2)

We also define the chirp parameter D as the separation between ${\lambda }^{+}$ and ${\lambda }^{-}$:

$$D={\lambda }^{+}-{\lambda }^{-}$$

(3)

The wavelength shift $({\lambda }_s-{\lambda }_0)$ determines the state of the sensor. The sensing signal S is a function of D and the FBGS parameters, such as its length L and its reflectivity, which is proportional to its coupling constant $\kappa$.

A simple expression for S can be obtained using the approximation that the FBG spectrum has a Gaussian shape, and the wavelength separation D is small. For a spectrum with a $1/e$ bandwidth w, S is just the derivative of the Gaussian divided by the Gaussian function itself, and we find:

$$S=-2\frac{D}{w^2}({\lambda }_s-{\lambda }_0)$$

(4)

S is then a linear function of $({\lambda }_s-{\lambda }_0)$, with a slope that is proportional to D and inversely proportional to the square of the bandwidth w. One of the main characteristics of DWDD is that the range of the sensor measurement is limited by the bandwidth of the FBGS. While in the case of a Gaussian spectrum, S is linear and can in principle take values from $-\infty$ to ∞, for too large a value of $({\lambda }_s-{\lambda }_0)$, the reflected power is too small and the signal gets lost in the noise. Thus, the measurement range is some multiple of w, determined by the signal-to-noise ratio (SNR) of the detection system. On the other hand, the sensitivity is proportional to the slope of S. An important quantity to characterize a given sensor system is the ratio of the range of the sensor to its resolution, or most commonly its base-2 logarithm B. The SNR can also be expressed in bits as $SN{R}_b={log}_2(P_0/P_n$), where $P_n$ is the noise-equivalent power. The inverse of that ratio is also often expressed as a percentage of the full scale % FS. Considering that the smallest change in S that can be detected for a given amount of noise is inversely proportional to the SNR, and taking the range as some multiple of w and the resolution as the inverse of the slope of S, we obtain:

$$B\propto SN{R}_b-{log}_2\left(\frac{w}{D}\right)$$

(5)

Thus, B is always less than $SN{R}_b$, but for a given SNR, B can be kept the same as long as the ratio $(w/D)$ is the same. The measurement range itself can be made larger with a shorter FBGS, and/or a higher reflectivity, both of which increase w. Equation (5) gives a simplified view, however, and only provides a qualitative understanding of the main feature of DWDD, which is that the bit resolution B and the measurement range can be optimized with a proper sensor design as we will show in Section 4.

For an actual implementation, which we have used in our demonstration system, it is more practical to use FBGS made from a uniform fiber Bragg grating of a given length. As we will show below, this is also beneficial to maximize B. Furthermore, the spectrum of a uniform FBG is described by a well-known analytical function, which makes both the analysis of their performance and their calibration easier.

Considering then the general case of a non-Gaussian FBGS spectrum, we can derive a general formula for the resolution $\Delta {\lambda }_{min}$, defined as the smallest change in sensor wavelength that can be detected. We consider that the measured power has a certain amount of random noise-equivalent power $P_n$, and we look for the smallest change in ${\lambda }_s$ that can be detected. From Equation (1), we obtain:

$$S({\lambda }_s-{\lambda }_0+\Delta {\lambda }_{min})-S({\lambda }_s-{\lambda }_0)=\left(\frac{\partial S}{\partial \lambda }\right)\Delta {\lambda }_{min}$$

(6)

and:

$$\left(\frac{\partial S}{\partial \lambda }\right)\Delta {\lambda }_{min}=\left[\frac{P^{+}+P_n-P^{-}+P_n}{P^{+}+P^{-}+P_n+P_n}\right]-\left[\frac{P^{+}-P^{-}}{P^{+}+P^{-}}\right],$$

(7)

where $P_n$ is taken as always positive and additive. Considering that the reflected power is the product of the incident power $P_0$ by the grating reflectivity $R\left(\lambda \right)$, we obtain the following formula:

$$\Delta {\lambda }_{min}=\frac{2}{\left(\partial S/\partial \lambda \right)}\left(\frac{1}{SNR}\right)\frac{1+\left|S\right|}{\left(R\left({\lambda }^{+}\right)+R\left({\lambda }^{-}\right)\right)}$$

(8)

where $SNR=P_0/P_n$. Armed with this formula, we can now find the best design to maximize the resolution for a given measurement range. But first, we review previous implementations of DWDD, and describe our current one.

3. Implementation

The first demonstration of DWDD using the algorithm of Equation (1) used nanosecond pulses from a single DFB laser in conjunction in a TDM arrangement combined with SDM [14,16]. For TDM-SDM, the launched pulse was divided by a splitter, and an incremental delay is added to each output branch so that reflected pulses arrive at different times onto a single photo-detector. A fast analog-to-digital (ADC) converter (200 Ms/s) was then used to digitize the reflected signal, and a similarly fast FPGA implemented the DWDD algorithm in real time. The pulse frequency was as high as 2 MHz. Despite the 10 bit resolution of the ADC, oversampling over more than 1 million pulses per second increased the effective resolution. The chirp D was obtained in two ways. One was to launch two successive pulses with different driving currents, taking advantage of the so-called “adiabatic chirp”, which is the instantaneous dependence of the DFB laser wavelength on the driving current. However, it was also found that within the pulse itself, the wavelength goes through a dynamic chirp [21,22] so that the wavelength at the end of the pulse is different from that at the beginning. Thus, we could use time samples at the beginning and end of a single pulse to measure $P^{+}$ and $P^{-}$. The resulting D was found to be equivalent for the two cases, being up to 110 pm. For this implementation, very low reflectivity gratings were used (about 1.5%), and S was an almost linear function, with an $r^2=0.998$ over a range of 40 °C. A precision of ±0.16 °C was obtained from the linear fit, which could be increased to ±0.13 °C by using a third-order polynomial fit. The resolution was estimated at 0.056 °C, or 0.56 pm in terms of wavelength.

The use of SDM with TDM has the advantage that a single photo-detector can be used for all the sensors. However, this results in a signal that decreases with the square of the number of branches N. A pure SDM scheme can otherwise be used, where each sensor has its own photo-detector, but still using a single laser source. Since the most expensive part of the system is the DFB laser, such an implementation is preferable, as the reflected signal is proportional to $P_0/2N$ instead of $P_0/N^2$. As we will show, a very large SNR can be obtained for 8 or more sensors.

The SDM implementation has the further advantage that short pulses are not required, which means that much higher-resolution ADC (16 to 18 bits) can be used. Finally, the chirp D can be much higher so that a larger measurement range is available from a combination of larger SNR and wider bandwidth.

A first implementation of such a low-speed SDM scheme was demonstrated for the measurement of tilted fiber Bragg grating sensors (TFBGSs) [15,17,18]. In such a case, SDM was required because the TFBGS is measured in transmission. For that implementation, the DFB was modulated with a 5 kHz square wave. By measuring the shape of the transmitted signal for different laser temperatures (and therefore different wavelengths), the adiabatic and dynamic chirps could be measured and fitted with a model involving three time constants in addition to the adiabatic instantaneous chirp, similar to what was found in ref. [21]. Whereas a chirp of only up to about 110 pm could be obtained on the nanosecond time scale of our previous experiment [16], the higher chirp coefficients associated with the longer time scales promise much larger values of D. In Ref. [15], only a $D=60$ pm was used by keeping the modulation amplitude small, as the bandwidth of the sensors itself was small (172 pm), due to their 10 mm length.

In the first demonstration of that system [15], a refractive index resolution of $9.9\times {10}^{-6}$ was obtained. With further improvements, such as using PM fiber, and a reference sensor to compensate for laser and ambient temperature changes, the refractive index limit of detection (LOD) was reduced to $3\times {10}^{-6}$. In terms of wavelength, this corresponds to a 40 fm resolution, more than 50 times that of a typical OSA. Using the same system, a LOD of 11 ppb for dissolved ammonia was demonstrated [18], again outperforming both expensive commercial systems, and other fiber optic implementations [23].

The main takeaway from those initial implementations is that the DFB laser is a very precise source. Because of its high power and high spectral density, a very high SNR is achievable, which results in very high resolutions that are ultimately limited mostly by the stability of the DFB wavelength and its dynamic chirp. Those can be compensated for by the use of two reference sensors as first proposed in ref. [17]. We elaborate on that technique in Section 5.

Following those implementations, we have started work on a similar SDM scheme, using the same 5 kHz square wave modulation but for measuring FBGS in reflection instead of TFBGS in transmission. We present here some initial results on the characterization of that system.

The setup is illustrated in Figure 2. A laser driver and temperature controller (Thorlabs CLD1015) was used to drive a DFB laser (SWLD series, Allwave Lasers), which had a nominal power of 10 mW at 80 mA current, and a wavelength of 1550 nm. The square wave modulation was fed to the driver from the ADC module (Measurement Computing model USB1808-X), which had 8 input channels and 2 analog output channels The ADC had 18 bits of resolution and 200 ks/s digitization rate for each channel. The output from the DFB output was split into 8 branches by a combination of a $1\times 2$ and two $1\times 4$ splitters (schematized as a single $1\times 8$ in the figure). Each branch then had a $2\times 1$ splitter, and the second branch of each, collecting the reflected signals, was sent to an amplified photo-detector (APD, Thorlabs model PDA20CS2). Only three such photo-detectors were available, so up to 3 sensors could be used.

The use of the DFB laser at such a low frequency means that its coherence length can be very long, compared with that of a nanosecond pulse. Care must therefore be taken to eliminate spurious reflections that can interfere with the signal and cause unwanted noise. For this, not only was the sensor fiber end cleaved at an angle but good quality components were used, such as couplers and connectors, that have >50 dB return loss. The low level of noise in our measurements is indicative of a negligible effect of such interference.

Figure 2b shows the design of the FBGS, meant for temperature sensing. The FBG is only about 0.67 mm long and is written within a few 100 μm of the angle cleaved end of the optical fiber. The angle of the cleave is about 15 degrees, which should result in <−50 dB back reflection. This design makes for a very compact sensor, with an active area that is less than 1 mm long, similar in size to typical thermistors but also to competing fiber optic temperature sensors such as GaAs devices.

A dry block temperature calibrator (East Tester ET2501) was used to test and calibrate the sensors over a range of temperature from −20 to 120 °C. For this, the sensors were affixed on the long probe of a precision USB thermometer (Thermoworks). The calibration bath was also used at fixed temperatures to measure the stability of the sensors.

The dynamic chirp model that was extracted from the measurements in ref. [15] can be used to estimate what value of D can be achieved. Using the same parameters, Figure 3 shows that for 80 mA of modulation amplitude, and by proper adjustment of the first measurement sample, a chirp D of up to 700 pm can be obtained. The red dots in the figure are the time samples of the ADC separated by 5 μs. The large dots correspond to the samples used in our experiments: a single one for $P^{-}$ at the beginning of the modulation state, and five averaged samples for $P^{+}$ at the end. By adjusting the trigger, the time of the first sample was set to be about 1 μs after the onset of the high-modulation state. The signal S was then averaged over 1000 cycles, and a further running average of 10 measurements was taken for an effective response time of 2 s.

The value of D for a given modulation amplitude and bias current is readily measured by scanning the DFB laser temperature, and measuring the reflected power $P^{+}$ and $P^{-}$ as a function of the wavelength, using the wavelength dependence on temperature, which we found to be 100 pm/°C. This measurement is faster than scanning the temperature of the dry block calibrator, which takes many minutes to settle for each temperature change. Measuring $P^{+}$ and $P^{-}$ as well as S as a function of the laser wavelength, we obtained the results shown in Figure 4a, where the top graph shows the powers and the middle graph shows the value of S obtained from them. For this graph, and the rest of the paper, we multiply S by a factor of 10,000 since in that case the best resolution is about 1 unit. Note that the S curve is now highly nonlinear as opposed to that obtained from a low-reflectivity, quasi-Gaussian FBGS in ref. [16]. The shape of the reflected spectrum is fitted to the well-known response of the reflectivity R of a uniform fiber Bragg grating:

$$R=\frac{sinh\left(\sqrt{{\kappa }^2-{\delta }^2}L\right)}{{cosh}^2\left(\sqrt{{\kappa }^2-{\delta }^2}L\right)-({\delta }^2/{\kappa }^2)}$$

(9)

where $\kappa$ is the coupling constant, L is the length, and:

$$\delta =2\pi n_{eff}\left(\frac{1}{{\lambda }_s}-\frac{1}{{\lambda }_0}\right)$$

(10)

and $n_{eff}$ is the effective refractive index of the optical fiber. The best fit in this case is for $\kappa =0.79{\mathrm{mm}}^{-1}$, $L=0.66$ mm, and $D=550$ pm. Using larger modulation amplitude, a value of D up to 720 pm is obtained.

As can be seen in Figure 4a (middle graph), the useful range of S is from about −8000 to 8000, after which either $P^{+}$ or $P^{-}$ becomes too small.

Using two similar sensors kept at a constant temperature in the dry block calibrator, with the temperature of the DFB laser fixed, we could estimate the measurement noise, and the SNR via Equation (8). Since the two sensors have slightly different reflectivity and bandwidth, we need to normalize their signal in order to quantify their difference in units of temperature. To do this, we introduce the normalized quantity $\overline{T}$:

$$\overline{T}=\frac{S}{\delta S/\delta T}$$

(11)

where the derivative is taken at the measurement temperature. We actually measured it at that point by scanning the laser diode temperature over a small temperature interval of typically $\pm 0.01$ °C, taking measurements at five temperatures separated by 0.005 °C, and obtaining the slope from a linear fit. The derivative is slightly different for each sensor, and is also a function of S. The normalized $\overline{T}$ is in units of temperature, and for small temperature change, its derivative is unity. Therefore, the change in $\overline{T}$ when everything else is kept constant is either a small fluctuation of the ambient temperature or that of the DFB, as long as the temperature does not vary too much. In such a case, the measurements from the two sensors should track each other, and their difference represents the measurement noise.

Figure 5 shows the measurement of $\overline{T}\left(t\right)-\overline{T}\left(t_0\right)$ for the two sensors, where $t_0$ is the starting time, to which is added the nominal temperature of the calibrator (25 °C), over 2200 s. The temperature of the DFB laser is set so that the value of S for both sensors is around −4000. The two sensors are seen to track each other very well, and both drift in the same way, for a combination of the reasons cited above. The difference between the two is shown in the bottom graph. There appears to be a fast noise component, as well as a larger slower fluctuation. The standard deviation is 0.005 °C. Other similar measurements are consistently below 0.006 °C. Using the parameters of the FBGS and the value of D from the fit of Figure 4, we can adjust the SNR in Equation (8) and we plot it in the bottom graph of Figure 4 so that the resolution at $S=-4000$ is 0.006 °C. The resulting $SN{R}_b$ is 15.8 bits.

The conclusion from those first measurements is twofold. One is that a high SNR is readily obtained, which means that high bit resolutions can be reached. Another is that the drift of the laser wavelength and/or dynamic chirp has to be eliminated or corrected in order to take advantage of the high resolution, which can be performed with reference sensors. In the next section, we show how we can design the sensor to maximize the resolution for a given measurement range. In particular, we target a resolution of more than 14 bits over a temperature range of 180 °C. We then address the question of calibration and real-time error correction through the use of two internal temperature-controlled sensors.

4. Optimization of the FBG Sensor Design

As per Equation (8), the resolution depends on the derivative of S, which in turn depends on the slope of the reflection spectrum. The function describing the spectrum of a uniform FBG as given by Equation (9) is known to approach a flat top spectrum for high values of the product $\kappa L$. Thus, the slope is typically smaller around the center (highest reflectivity) and larger on the edges, where it attains high values as the reflectivity approaches the first zero on either side. The resolution is therefore always smaller at the center. However, as the slope increases on the edges, by increasing the resolution, the reflectivity decreases, which reduces the resolution because of the lower SNR. In contrast, for a Gaussian-shaped spectrum, the slope is constant, but the reflectivity still decreases on the edges, and therefore the resolution is always a maximum at the center.

Figure 4b shows the temperature resolution for a FBGS of 0.65 mm length, and reflectivity of 61%. As expected, the resolution improves on each side of the center, where the slope of S increases before it degrades again rapidly due to the decreasing SNR. In the case of Figure 4a, the lower reflectivity results in an almost flat resolution that degrades rapidly near the edges. We will therefore define the useful range as that where the resolution is at least as good as the center as indicated with the dashed lines in the bottom graph of Figure 4b. The base-2 logarithm of the ratio of the range to the resolution at the center gives the bit resolution B. It can readily be seen that for a similar resolution, the range can be greatly expanded by having a higher reflectivity.

The question is, therefore, what is the best compromise in terms of grating reflectivity and length to maximize both the resolution and the range? We give here one example of an optimized design. We set a target measurement range of 180 °C for temperature measurement, and a resolution B of at least 14 bits, or 0.0061% FS. This corresponds to a minimum resolution of 0.011 °C but also exceeds the accuracy specification for a class C6 load cell, which is 0.0116% FS, or 13.1 bits. Class C6 load cells are currently considered to have the best accuracy. It is important to have a resolution higher than the targeted accuracy to leave room for multiple factors that can degrade both resolution and accuracy in the final manufactured product. Our intent is to show that the DWDD system can have performance meeting these demanding commercial specifications. We use Equation (8) with an $SN{R}_b$ of 15.8 bits, corresponding to our experimental estimation. Another constraint is that the FBGS has a coupling constant limited by the maximum achievable refractive index modulation $\delta n_{max}$. Although higher values have been routinely obtained, we choose $\delta n_{max}=1.2\times {10}^{-3}$ as the practical limit. This is because to achieve long-term stability, the FBG has to be strongly annealed at high temperature and in the process may lose as much as half of the original $\delta n$.

The first step is to determine the best choice of chirp D. Figure 6 shows the measurement range and B as a function of the FBGS peak reflectivity for lengths of 0.6, 0.65, and 0.7 mm. The curves for shorter FBG stop at smaller reflectivities due to the $\delta n$ limit. B is seen to have a broad maximum around a reflectivity of 65%. The measurement range is larger for smaller D, but B is then smaller than 14. For $D=550$ pm, a range of 180 °C and $B=14.4$ bits is achievable for $R=66\%$ and a length $L=0.6$ mm. A design with $D=500$ pm, $L=0.65$ mm, and $R=71\%$ gives a similar B.

Figure 7 looks at the best choice of length. For the same reflectivity, a shorter length requires a higher $\delta n$, so the curves are bounded on the short side by the limit $\delta n=1.2\times {10}^{-3}$. A smaller chirp D gives a larger range, but a lower B. Taking a chirp of 500 pm, and a length of 0.55 mm, one can obtain a range of 200 °C and still have $B>14$. In general, the higher the reflectivity, the larger the range. This may require a larger D, but as we have shown, D up to 700 pm can be achieved. If a larger $\delta n$ can be practically reached, then even larger measurement ranges can be obtained with shorter FBGS.

5. Calibration

All sensor systems require proper calibration. A sensor ecosystem is one where both sensors and instruments have standardized performance and calibration and can be used interchangeably so that users can rely on the documented performance. Sensor and instrument calibration can only be performed with respect to other sensors, measured under controlled conditions. Ideally, the calibration will be traced to international standards. While this seems obvious, the chain of sensors, instruments, and measurements that can establish that link is complex and subtle. Furthermore, there is the issue of long-term calibration: once an instrument or a sensor has been calibrated at one point in time, how can it be ensured that measurements made at future times still retain their validity?

Sensor and instrument calibration is not an issue that has been generally much discussed for FBGSs [24,25]. Since many systems are application- and user-specific, calibration is often just a crude value of either temperature or strain sensitivity without any traceability, and users are often left to perform the calibration themselves. The DWDD technique, as we will try to demonstrate, offers the potential to use highly standardized sensors and instruments. We explain here how a calibration of both sensors and instruments can be made that includes traceability to international standards. Furthermore, by exploiting the multichannel nature of the DWDD instrument, calibration can be maintained by real-time comparison and correction using internal reference sensors kept at constant temperature, and the long-term stability of the calibration only depends on the relative long-term stability of a single temperature sensor in that reference cell. Thus, if the centralized calibration is performed under conditions traceable to international standards, that traceability extends to all instruments and sensors with quantifiable accuracy, stability and reliability, guaranteed in real time for every measurement. This is a property that is actually rarely found with other sensing technologies, where it usually only comes at a much higher price, and it makes the DWDD ecosystem a very appealing proposition.

The response of a DWDD sensor is, contrary to that of a usual FBGS, related to the spectral shape of the sensor via Equation (1). A uniform FBG is characterized by only three parameters: the length, the coupling constant, and the central wavelength for given environmental conditions (temperature and strain). Knowing those parameters with enough precision allows easy computing of the function S but also its local derivatives with respect to wavelength or chirp. For a given sensor, further calibration requires knowledge of the sensitivity to the parameter to be measured, for example, temperature or strain. For this last step, calibration methods already used for similar sensors made with other technologies can be used, such as the dry bath calibrator used above. Sensors can be individually calibrated for the best precision, or can use standard measured values of temperature or strain sensitivity. For example, thermistors are characterized by a well-known response function, the coefficients of which are supplied for each sensor. Load cells are characterized by a gauge factor, temperature sensitivity and so on. While in early days, the computing of complex response functions involving many parameters was tedious, and lookup tables were often used, this problem does not exist any more. Calibration parameters can be stored in instrument memory and in the sensor connector itself, with computation performed in real time.

As the calibration relies on computing parameters from the spectral shape of the FBG, a first question is whether the FBG can be made to actually fit the theoretical shape given by Equation (9). Figure 8 shows the spectrum on a semi-log scale of one FBG measured with an OSA along with its best fit. The sum of the least squares indicates an average deviation of 0.16 % from the theoretical formula for the region of the central lobe typically used for the sensor operation, and the plotted spectra are visually indistinguishable. The accuracy may be limited by that of the OSA. The side lobes of the measured spectrum are higher than those of the theoretical one, save for the first one on the right, for reasons unknown. Otherwise, the spectrum is perfectly symmetric. The other sensors we used showed similarly good quality spectra. FBG can be fabricated repeatedly with the same exact length as defined by a hard aperture in front of the fiber, for example. The only adjustable parameters are then the coupling constant and the central wavelength, that can nevertheless generally be controlled within 1% and 50 pm, respectively.

We have shown that the DWDD instrument has very high resolution but is subject to fluctuations and drift of the DFB laser wavelength and dynamic chirp. Let us first assume that the instrument is nevertheless stable and discuss the sensor calibration, with the example of a temperature sensor. The FBGS can first be calibrated spectrally with the highest possible accuracy, for example, using a wavelength-swept source in conjunction with an optical wave meter that typically has a guaranteed accuracy at the sub-pm level. Knowing the length with similar precision, the coupling constant can then be evaluated.

To calibrate the temperature response, a dry bath calibrator can be used. Scanning the temperature over the desired temperature range, values of S are recorded at a certain number of points. The resulting data are then fitted to obtain the temperature dependence.

Extracting the value of temperature from the measured value of S means inverting this process. However, the FBG response of Equation (9) does not have a simple analytic inverse function. While the inverse can be computed numerically, another choice is to build a lookup array using Equation (9) and the temperature dependence, and interpolate the value of T from the measured value of S. This is the approach we take in our simulations.

Figure 9 shows an attempt at such a calibration, with two sensors being measured at the same time. For this, we did not characterize the spectral shape with the accuracy described above but performed a nonlinear multiparameter fit of the S data using Equation (9) with both the FBG parameters and temperature dependence as independent parameters, as well as the chirp D. Since the two sensors were measured simultaneously, we constrained the value of D to be the same for both, as well as their temperature coefficient. The length was allowed to vary between the two sensors only by a few μm. As can be seen in Figure 9a, the two sensors have slightly different central wavelengths but similar S response because their design is otherwise identical, with nearly the same length but slightly different reflectivity. From the best fit parameters, an array of 180 values of S as a function of temperature was generated. Using this array, for all measured values of S, the value of T was then calculated back by spline interpolation from the lookup table. Doing so with the two sensors, we then calculated the difference between the temperature at which the original S values were measured with the temperature obtained from this reverse operation. The result is shown in Figure 9b. What we found is that the maximum deviation was −0.8 °C, but in fact the two sensors still tracked each other very well as in Figure 5. Thus, the standard deviation of their difference (green line) is 0.055 °C. The 0.8 °C drift can thus be attributed to the instrument drift as is already observed in Figure 5. The calibration procedure took place over a few hours since the calibrator temperature took about 20 min to settle for each point. The relative drift between the two sensors is larger than the 0.005 °C difference of Figure 5, which is probably in part attributable to the relative drift in the polarization state of the light as will be discussed in Section 6.

The drift in the DFB wavelength and chirp therefore still appears to be the main limit to the instrument precision. In order to compensate for that drift, we discuss here the use of an internal reference cell (RC) comprising two sensors maintained at constant temperature as already proposed in ref. [17]. Two reference sensors are required because two parameters can drift: the wavelength and the chirp. The advantage of using FBGS themselves as references is that FBGS can be very stable in time, and are ten times less sensitive to temperature than DFB lasers. Since the FBGS internal reference has the same temperature dependence as the FBGS temperature sensors themselves, the measurement is as stable as the temperature of the RC. TEC controllers can easily reach 0.01 °C or better stability, with proper thermal and mechanical design, and FBGS have very low volume, so the thermal uniformity between the sensors and the reference thermistor can be made very good. For example, in ref. [17], the 40 fm stability achieved with one reference sensor corresponds to 0.004 °C. It is to be noted that only the long-term relative accuracy of the reference thermistor is required and not absolute long-term accuracy. Thermistors with such long-term stability are commercially available.

Figure 10 illustrates how a signal correction using internal referencing can be performed with two sensors having central wavelengths shifted on both sides of a standard central wavelength (e.g., 1550 nm) by an amount such that the values $S_{r1}$ and $S_{r2}$ of the two reference sensors more or less match that of the best resolution as in Figure 4b (bottom graph). For the example in Figure 10, $S_{ri}\approx \pm 4500$. Assuming identical sensors, if the temperature of the DFB laser drifts, each $S_r$ curve will shift to the right or to the left, changing the $S_{ri}$ values ($i=1,2$) in the same direction for both as shown in Figure 10a. In the ideal calibrated state, the sum of $S_{r1}$ and $S_{r2}$ is set to be 0 (as one is positive and the other negative).

We can define a normalized calibrated wavelength ${\overline{\lambda }}^c$ by similarly dividing S by its derivative with respect to the wavelength, and taking the average of those ratios for both sensors. At the time of calibration, we have:

$${\overline{\lambda }}^c=\frac{1}{2}\left[\frac{S_{r1}^c}{\partial S_{r1}/\partial \lambda }+\frac{S_{r2}^c}{\partial S_{r2}/\partial \lambda }\right]$$

(12)

For two identical reference sensor, ${\overline{\lambda }}^c=0$, but its value may be different if the sensors are slightly different. The drift in the laser wavelength $\Delta \lambda$ can then be obtained by the difference between the normalized wavelength at the time of measurement ${\overline{\lambda }}^m$ and ${\overline{\lambda }}^c$

$$\Delta \lambda ={\overline{\lambda }}^m-{\overline{\lambda }}^c=\frac{1}{2}\left[\frac{S_{r1}^m-S_{r1}^c}{\partial S_{r1}/\partial \lambda }+\frac{S_{r2}^m-S_{r2}^c}{\partial S_{r2}/\partial \lambda }\right]$$

(13)

On the other hand, if the chirp D is drifting, the function S has the same zero but a different slope as shown by the simplified model of Equation (4). Therefore, the difference between the two S values will change. We can similarly define a normalized calibrated chirp value $\overline{D}c$:

$${\overline{D}}^c=\frac{1}{2}\left[\frac{S_{r1}^c}{\partial S_{r1}/\partial D}+\frac{S_{r2}^c}{\partial S_{r2}/\partial D}\right]$$

(14)

The drift in D can be obtained by:

$$\Delta D={\overline{D}}^m-{\overline{D}}^c=\frac{1}{2}\left[\frac{S_{r1}^m-S_{r1}^c}{\partial S_{r1}/\partial D}+\frac{S_{r2}^m-S_{r2}^c}{\partial S_{r2}/\partial D}\right]$$

(15)

Therefore, the two reference sensors are characterized by the partial derivative of their response with respect to a change in the laser wavelength and chirp. Those are defined as calibration factors for each instrument and are obtained from the measured FBG parameters for each sensor in conjunction with Equation (9). They can also be measured directly by scanning the DFB laser wavelength and the modulation amplitude over a small interval. As with any calibration procedure, the result will be only as good as the instruments used, and therefore, great care must be taken to perform those measurements with instruments that are themselves precisely calibrated.

The key to correct the measurement for the laser drift is to obtain the real value of S for the measured sensor as would be obtained with a well-calibrated instrument, where the wavelength of the DFB laser and its dynamic chirp are determined from the values of $S_{ri}c$ at the time of calibration. One way is to set the DFB laser temperature at the time of calibration such that:

$${\overline{\lambda }}^c=\frac{S_{r1}^c}{\partial S_{r1}/\partial \lambda }+\frac{S_{r2}^c}{\partial S_{r2}/\partial \lambda }=0$$

(16)

Similarly, the normalized chirp at time of calibration ${\overline{D}}^c$ is recorded. Thereafter, the measured value of a sensor $S^m$ can be corrected, also knowing the values of its partial derivatives around $S^m$, which are also obtained from its well-characterized FBG parameters, in order to obtain the calibrated value $S^c$:

$$S^c=S^m-\left[\frac{\partial S}{\partial \lambda }\Delta \lambda \right]-\left[\frac{\partial S}{\partial D}\Delta D\right]$$

(17)

From the corrected value $S^c$, the value of the measurand (e.g., temperature) is then obtained from the lookup calibration data by interpolation. This way, the measured value remains the same as at the time of calibration.

This technique has the further advantage that it can be extended to the use of centralized “golden sensors” (GSs) used to calibrate all instruments, as well as a centralized precisely calibrated “golden instrument” (GI). This way, all instruments are calibrated using the GS, and all sensors are calibrated using the GI, which is itself corrected in real time using the GS. This is made possible because of the intrinsic multichannel nature of the DWDD instrument that allows multiple sensors to be measured at the same time. Furthermore, if the GSs are maintained under conditions that are traceable to an international standard, then the traceability is transferred to all instruments. For example, the GS can be placed in a so-called fixed-point cell (FPC) that is kept at the fusion temperature of a given element, such as Ga, which is 29.7646 °C, and is an ITS-90 fixed point, thus an international standard.

The calibration procedure is illustrated in Figure 11. The first step (Figure 11a) is to set the wavelength and dynamic chirp of the GI to values that will then be fixed as the standard values to be reproduced in every other instrument. For this, those values are set with respect to the values of S measured from the two GSs. The value of D can be measured as in Figure 4 and the modulation amplitude set so that D is approximately the desired standard value (e.g., 550 pm). The temperature of the DFB is also set such that the sum $S_{g1}+S_{g2}$ has a desired value, which can be 0, indicating that the center wavelength is approximately at the center between the two GSs. At that point, both the derivatives of S with respect to D and $\lambda$ can be measured, as well as calculated from the known spectral shape of both GSs. Thus, the normalized ${\overline{\lambda }}^c$ and ${\overline{D}}^c$ can be calculated. For any future measurement, the deviation from those calibrated values indicates a drift of wavelength or chirp and is used for correcting the measurements made by the GI. Those values are also taken as standard values for the sensor ecosystem.

The next step is to calibrate another instrument using the GS (Figure 11b). The instrument under calibration (IUC) is connected to the two GSs, and the two reference sensors for that instrument are also placed in the FPC. The laser temperature and modulation amplitude are then set such that $\overline{\lambda }$ and $\overline{D}$ from the GS are equal to the standard values ${\overline{\lambda }}^c$ and ${\overline{D}}^c$. This means that the wavelength and chirp of the IUC are the same as those of the GI. At the same time, the values of ${\overline{\lambda }}^r$ and ${\overline{D}}^r$ from the reference sensors (RS) are measured. Those become calibration factors for the instrument.

The last step (Figure 11c) is to place the RS into the temperature-controlled reference cell (RC), connected to the IUC, while the two GSs are still connected to two other ports the IUC. Now the temperature of the RC is adjusted so that the values ${\overline{\lambda }}^r$ and ${\overline{D}}^r$ are equal to the calibrated values when the reference sensors are in the FPC. This would normally mean that the temperature of the RC is equal to that of the FPC.

The GI can be made with superior specifications to usual instruments, for example with a higher power laser and a high-performance temperature control. It can also be kept under stable environmental conditions. As will be discussed below, it can also use all polarization-maintaining (PM) components, and the GSs themselves can be made in PM fiber to eliminate any polarization dependence.

Finally, for sensor calibration (Figure 11d), a temperature calibrator can be used to measure them at multiple temperatures using the GI, which is corrected in real time because it is connected to the GS.

Now, all sensor measurements can be corrected to obtain the standardized calibrated value $S^c$ as would be obtained if the sensor were measured with the GI using the GS and reference sensors:

$$S^c=S_s^m-\left[\frac{\partial S}{\partial \lambda }\Delta \lambda \right]-\left[\frac{\partial S}{\partial D}\Delta D\right]$$

(18)

In order to test the robustness of the correction against expected fluctuations of the DFB laser, we performed a simulation including two GS and two RS with slightly different parameters. We calculated the temperature error for fluctuations of D up to 0.3%, and DFB temperature up to 0.05 °C, noting the maximum error for all combinations of those extreme drifts.

Figure 12a shows the maximum and minimum values of the measured temperature without the correction factor. Figure 12b shows the same variation with the correction factor. The error is reduced from ±0.6 °C to less than ±0.005 °C over the entire measurement range. Therefore, the accuracy is ultimately as good as the resolution itself.

However, in order to achieve this level of precision, the sensors and the reference FBGS must be characterized with that level of accuracy. The FBGS shape is determined by two parameters: the product $\kappa L$ and the length L. Figure 12c shows that if the value of $\kappa L$ differs by 0.1% from its real value, the error increases to 0.046 °C. This is nevertheless still only 0.026% of the full range, and is still a more than ten times improvement over the error without the correction. Furthermore, the parameters of the FBGS can be further adjusted by checking and correcting the calibration at two extreme locations within the measurement range. As for the reference sensors, their calibration needs to be accurate only around their point of operation. This can be accomplished as long as the system’s resolution is better than the required accuracy, which is the case here.

One important point is that once an instrument is calibrated relative to the GI, the stability of the calibration depends only on the relative stability of the reference cell. Therefore, the absolute calibration of the thermistor used to control the temperature of the RC is not important, only its long-term stability. It can therefore be said that the absolute accuracy of the GS as measured by the GI is transferred to the instrument, which remains calibrated within the long-term stability of the internal thermistor.

Finally, as the values $\overline{\lambda }$ and $\overline{D}$ are constantly monitored, if their value exceeds some limit, the instrument can easily be recalibrated by re-adjusting the laser temperature or its modulation amplitude to reset those values to ${\overline{\lambda }}^c$ and ${\overline{D}}^c$.

6. The Effect of Sensor Birefringence

The effect of sensor birefringence has also been rarely discussed, and yet it is the limiting factor in the system’s performance. The error on the measurement is directly proportional to the birefringence of the sensor because the DFB laser source is polarized. A birefringence of ${10}^{-6}$ gives roughly an error of 1 pm, translating to 0.1 °C, or $0.06\%$ FS. Good-quality single-mode fiber has a birefringence in the low ${10}^{-7}$, but the FBG writing process is known to create birefringence, though much of it can disappear upon high-temperature annealing. In any case, the amount of photo-induced birefringence is highly dependent on the FBG writing process.

For the sensors we used in our tests, by adding a polarization controller at the output of the DFB laser, we found that the temperature measurement could vary by about 0.1 °C, indicating a total birefringence of about ${10}^{-6}$. In such case, the long-term drift of the sensor measurement can happen if the polarization state of the light changes over time, which can easily happen if the pigtail is moved, bent, or twisted. The relative drift between sensors that we observed is therefore most probably due to that effect.

There are three solutions to overcome that problem. One is to have a writing process that minimizes the photoinduced birefringence in the FBGS. In that case, the measurement is still limited by the fiber intrinsic birefringence, which can vary from device to device. Another solution is to depolarize the light from the DFB laser. Commercial narrowband depolarizers, suited for our system, can reduce the polarization of the light by more than 95%. This has the potential to reduce the polarization sensitivity to less than 0.005 °C, smaller than the resolution of 0.006 °C.

The ultimate solution is of course to have a system built entirely with polarization-maintaining (PM) fiber. While it definitely adds to the cost, it would allow the potential of the system in terms of resolution and accuracy to be fully exploited. In fact, the GI and GS could themselves be made of PM components, thus guaranteeing their full accuracy. For many applications, however, an accuracy equivalent to ±0.05 °C, or 0.03% FS, is more than sufficient, and sensors with a low enough total birefringence would be suitable.

7. Conclusions

This preliminary work has shown that a FBGS sensor/instrument ecosystem based on DWDD is within the current state of technology, combining high resolution (14 bits or higher) and high accuracy, as well as traceable calibration. Furthermore, with its simplicity and use of standard components, the DWDD instrument can be made at lower cost than current WDM FBGS interrogators. While our measurements were made by splitting the laser into eight branches, there was still plenty of power for at least another doubling of channels or more with a more powerful laser or more detector gain. Keeping two channels for the reference cell, this still makes instruments possible with up to 14 or 30 channels. Since a large fraction of the cost is in the DFB laser and the ADC electronics, the cost per sensor is less and less for such high-channel count instruments. Another feature of the system is that FBGS can measure multiple parameters. Keeping the core FBGS design the same, the strain, temperature, pressure, and other sensors could all be measured by the same instrument with the same bit resolution, and all these sensors retain the advantages of being optical, such as immunity from electromagnetic interference, but also the possibility of having very long lead cables. This is of course a feature of all FBGS systems but not of systems based on electrical sensors. A standardized sensor ecosystem based on DWDD opens the way to such universal instruments that could accept multiple sensor types in a “plug and play” fashion.

On top of being standardized, the DWDD-FBGS sensors are much shorter than typical FBGSs, being only a fraction of a millimeter long. This greatly facilitates their packaging or use in constrained areas. For example, the probability of failure when a length of optical fiber is stretched is proportional to its length, so a 0.6 mm long FBGS is more than ten times less likely to fail under strain than a typical 8 mm long one. For temperature sensing, this also results in a much more local measurement. As we have shown, large measurement ranges are possible for the same bit resolution by using even shorter FBGS, limited only by the maximum refractive index modulation of the FBG. Sacrificing some of the bit resolution also allows extending the range as is clear from Figure 6. Most applications require precision measurement over a limited range. For load cells, and other sensors based on strain measurement, such as pressure sensors and accelerometers, the mechanical construction determines the range. For temperature, extending the range involves other problems, such as high-temperature coatings and the robustness of the FBG. Common applications such as hot spot transformer monitoring typically do not involve temperatures higher than 150 °C. The advantage of the DWDD system is that it can keep the same bit resolution B for a different range, contrary to WDM-type FBGS systems.

The creation of a standardized sensor ecosystem can bring large economies of scale from volume production. The standard design of the FBGS can result in both low-cost manufacturing, and consistent performance through a high degree of automation. The DFB laser already benefits from dramatically reduced cost and compact packaging due to its pervasive use in small form-factor pluggable (SFP) transceivers. The optical part of the instrument (splitter and couplers) can be integrated onto a single optical chip. The signal processing can even be integrated in a single IC chip that performs multichannel trans-impedance amplification, analog-to-digital conversion, and computation of the S-signals. The technology to achieve such a high degree of integration at low cost is widely available today. Thus, the cost per sensor for a multichannel instrument can easily be competitive with the current electronic sensing technology for the same level of performance, while adding all the benefits of FBGS, such as compactness, plug-and-play multiparameter sensing, immunity to interference, and long lead cables.

Our future work is focused on developing a full prototype, and pushing the limits of resolution and accuracy to their full potential. However, the simplicity, low cost, and high performance of even a basic DWDD setup should allow other research groups to investigate more applications. In time, we believe that DWDD has the potential to replace traditional electrical-based sensing technologies, especially in high-precision or high-resolution large channel-count applications.