Performances of PMMA-Based Optical Fiber Bragg Grating Sensor in Extended Temperature Range
Performances of PMMA-Based Optical Fiber Bragg Grating Sensor in Extended Temperature Range
Zhang, Wei;Webb, David J.
2021-05-23 00:00:00
hv photonics Article Performances of PMMA-Based Optical Fiber Bragg Grating Sensor in Extended Temperature Range 1 , 2 , 3 Wei Zhang * and David J. Webb School of Aerospace, Transport, and Manufacturing, Cranfield University, Cranfield MK43 0AL, UK School of Optoelectronic Engineering, Qilu University of Technology, Jinan 250353, China Aston Institute of Photonic Technology, Aston University, Birmingham B4 7ET, UK; d.j.webbl@aston.ac.uk * Correspondence: zhang.wei@cranfield.ac.uk Abstract: PMMA based optical fiber Bragg grating (POFBG) sensors are investigated in an environ- mental chamber with controlled temperature and relative humidity at temperature extended to 70 C. At below a critical temperature of 50 C the POFBG sensor exhibits good linearity and sensitivity for both temperature and humidity sensing. Nonlinear responses are observed at higher temperature, giving rise to varying, reduced magnitudes of sensitivities. An important feature of POFBG humidity sensing is observed at above critical temperature where the POFBG humidity sensitivity turns from positive to negative. A theoretical model based on Lorentz–Lorenz equation is presented to estimate the dependence of POFBG refractive index on temperature and relative humidity. The experimental results qualitatively agree with the theoretical analyses. Keywords: fiber Bragg gratings; polymer optical fiber; thermo-optic effect; refractive index humid- ity dependence 1. Introduction Citation: Zhang, W.; Webb, D.J. Polymer optical fibers (POFs) are made of low-cost plastic materials, for example, Performances of PMMA-Based PMMA. Since POFs are considered as having high optical attenuation compared to their Optical Fiber Bragg Grating Sensor in silica counterpart, they have long been overshadowed by the success of silica optical fibers. Extended Temperature Range. Recent technological advancements have made POF networks competitive over a range of Photonics 2021, 8, 180. https://doi. important applications for short-distance data communication [1]. The physical and chemi- org/10.3390/photonics8060180 cal properties of polymeric materials are rather different to silica, potentially making them attractive for researchers to exploit in device and sensing applications. Bragg gratings have Received: 11 May 2021 been inscribed into step index and microstructured POF based on PMMA. The interesting Accepted: 21 May 2021 features of POFBGs include the negative refractive index (RI) change against temperature Published: 23 May 2021 rise and affinity for water that leads to a swelling of the fiber and an increase of RI. The former feature offers a well-conditioned performance for overcoming the cross-sensitivity Publisher’s Note: MDPI stays neutral issues existing in silica fiber while the latter feature leads to a humidity sensor in which the with regard to jurisdictional claims in Bragg wavelength of a POFBG increases with humidity [2]. POFBG has been successfully published maps and institutional affil- used as a humidity sensor and as a moisture sensor [3,4]. This is a potentially very useful iations. property, which has possible applications in the chemical processing, agricultural, food storage, paper manufacturing, semiconductor, and pharmaceutical industries [5]. POFBG sensors generally operate at around room temperature. POFs may suffer undesirable changes in their optical, thermal, and mechanical properties when aging Copyright: © 2021 by the authors. under high temperature. Note that the high temperature here is only referred to polymer Licensee MDPI, Basel, Switzerland. material performance. Conventional PMMA POFs are typically working up to 85 C [6]. This article is an open access article There has been no report on POFBGs operating up to that temperature. In this work distributed under the terms and conditions of the Creative Commons we investigate POFBG sensor performance in an extended temperature range in both Attribution (CC BY) license (https:// experiment and theory. creativecommons.org/licenses/by/ 4.0/). Photonics 2021, 8, 180. https://doi.org/10.3390/photonics8060180 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 180 2 of 8 Photonics 2021, 8, x FOR PEER REVIEW 2 of 8 2. Experiments 2. Experiments All the POFBGs used in the work are made of PMMA based step index optical fiber. All the POFBGs used in the work are made of PMMA based step index optical fiber. Before grating fabrication, the POF was annealed in an oven at 80 C over 7 h. POFBGs Before grating fabrication, the POF was annealed in an oven at 80 °C over 7 hours. POFBGs mentioned hereafter operate in the 1.5 m region. These POFBGs are usually UV-glued to mentioned hereafter operate in the 1.5 μm region. These POFBGs are usually UV-glued to a silica optical fiber lead to avoid the high optical loss in the polymer optical fiber [5]. The a silica optical fiber lead to avoid the high optical loss in the polymer optical fiber [5]. The POFBG used in the experiments is fabricated by attaching a 10 cm length of PMMA optical POFBG used in the experiments is fabricated by attaching a 10 cm length of PMMA optical fiber to a single mode silica fiber down-lead using UV curable glue (AT9390, NTT). This fiber to a single mode silica fiber down-lead using UV curable glue (AT9390, NTT). This UV glue has a transition temperature of 121 C, which is close to the PMMA transition UV glue has a transition temperature of 121 °C, which is close to the PMMA transition temperature. Although connectorized POFBG has been proposed [7], the POFs we used temperature. Although connectorized POFBG has been proposed [7], the POFs we used in the experiments are lab-made with considerably varying diameters [8], which makes it in the experiments are lab-made with considerably varying diameters [8], which makes it not suitable to be connectorized with standard single mode silica optical fiber for practical not suitable to be connectorized with standard single mode silica optical fiber for practical application. The UV-gluing technique thus was used to achieve stable operation. The application. The UV-gluing technique thus was used to achieve stable operation. The PMMA-based POF contained a 5 mm long FBG, fabricated by illuminating from above a PMMA-based POF contained a 5 mm long FBG, fabricated by illuminating from above a phase mask placed on top of the POF using 325 nm UV light from a HeCd laser. phase mask placed on top of the POF using 325 nm UV light from a HeCd laser. In the experiments, the POFBGs were placed inside an environmental chamber (Sanyo In the experiments, the POFBGs were placed inside an environmental chamber Gallenkamp) to operate at the desired temperature and humidity. They were illuminated (Sanyo Gallenkamp) to operate at the desired temperature and humidity. They were illu- via a fiber circulator with light from a broadband light source (Thorlab ASE730) and minated via a fiber circulator with light from a broadband light source (Thorlab ASE730) observed in reflection using an IBSEN I-MON 400 wavelength interrogation system, as and observed in reflection using an IBSEN I-MON 400 wavelength interrogation system, shown in Figure 1. as shown in Figure 1. Figure 1. Experimental setup used in this work. Figure 1. Experimental setup used in this work. POFBG Wavelength Responses in Extended Temperature Range POFBG Wavelength Responses in Extended Temperature Range For a specific temperature/humidity, the Bragg wavelength change of a POFBG For a specific temperature/humidity, the Bragg wavelength change of a POFBG against humidity/temperature change can be expressed as [9] against humidity/temperature change can be expressed as [9] ∆ = (+ )Δ Dl = l (h + b)D H B B (1) (1) ∆ = (+ )Δ Dl = l (a + x)DT B B where λB is the initial Bragg wavelength, η is the normalized refractive index (RI) change where l is the initial Bragg wavelength, h is the normalized refractive index (RI) change with humidity, β is the swelling coefficient related to humidity induced volumetric with humidity, b is the swelling coefficient related to humidity induced volumetric change, change, α is the thermal expansion coefficient (TEC), and ξ is the thermo-optic coefficient. a is the thermal expansion coefficient (TEC), and x is the thermo-optic coefficient. From (1), From (1), one can see that there are two factors contributing to the wavelength change of one can see that there are two factors contributing to the wavelength change of a POFBG. a POFBG. One is the RI change induced. The other is the length change of the PMMA- One is the RI change induced. The other is the length change of the PMMA-based fiber. based fiber. It has been reported [10] that solid drawn polymers and stretched elastomers ex- It has been reported [10] that solid drawn polymers and stretched elastomers exhibit hibit anisotropic expansion, which mainly depends on the polymer processing history. anisotropic expansion, which mainly depends on the polymer processing history. It was It was also noticed that rising temperature releases the residual drawing stress in the also noticed that rising temperature releases the residual drawing stress in the fiber, lead- fiber, leading to fiber shrinkage, thus causing an additional negative change of the POFBG ing to fiber shrinkage, thus causing an additional negative change of the POFBG wave- wavelength [11,12]. Annealing POFs can mitigate this issue. However, it has been reported length [11,12]. Annealing POFs can mitigate this issue. However, it has been reported that that the usual annealing process is far from enough [12,13]. This consequently gives rise to the usual annealing process is far from enough [12,13]. This consequently gives rise to inconsistent POFBG sensor performance [4]. inconsistent POFBG sensor performance [4]. An experiment was designed to examine the performance of the POFBG sensor. A pre-strain technique was used to eliminate the residual stress related inconsistency [14,15], Photonics 2021, 8, x FOR PEER REVIEW 3 of 8 Photonics 2021, 8, 180 3 of 8 An experiment was designed to examine the performance of the POFBG sensor. A pre-strain technique was used to eliminate the residual stress related inconsistency [14,15], in which the POFBG was strained using a translation stage and then glued to an INVAR in which the POFBG was strained using a translation stage and then glued to an INVAR bar. The device length change due to changing temperature in this case is not determined bar. The device length change due to changing temperature in this case is not determined by the POF thermal expansion but by INVAR thermal expansion (the influence of the glue by the POF thermal expansion but by INVAR thermal expansion (the influence of the glue is negligible as the glued points are very small in size compared to the POF length). In is negligible as the glued points are very small in size compared to the POF length). In addition, the fiber cannot lengthen in the longitudinal direction, but there can remain a addition, the fiber cannot lengthen in the longitudinal direction, but there can remain a deformation in the transverse plane. This consequently leads to the change of refractive deformation in the transverse plane. This consequently leads to the change of refractive index of the fiber which can be estimated by using the Lorentz–Lorenz relation [16]. This index of the fiber which can be estimated by using the Lorentz–Lorenz relation [16]. This change is insignificant due to the small ratio of fiber core diameter to fiber length and it is change is insignificant due to the small ratio of fiber core diameter to fiber length and it ignored in this work. The pre-strained POFBG was then placed in the environmental is ignored in this work. The pre-strained POFBG was then placed in the environmental chamber for the test of POFBG performance. Since the ends of the POFBG were fixed, the chamber for the test of POFBG performance. Since the ends of the POFBG were fixed, length of the PMMA optical fiber between the two clamping points does not vary with the length of the PMMA optical fiber between the two clamping points does not vary either temperature or humidity (given that the applied strain was larger than any temper- with either temperature or humidity (given that the applied strain was larger than any ature/humidity induced fiber length change). In this case, the POFBG temperature/hu- temperature/humidity induced fiber length change). In this case, the POFBG tempera- midity sensitivity only relies on the thermo-optic effect/RI humidity dependence of the ture/humidity sensitivity only relies on the thermo-optic effect/RI humidity dependence fiber. of the fiber. The highest operation temperature for the PMMA based fiber grating sensor, re- The highest operation temperature for the PMMA based fiber grating sensor, reported ported so far, was 50 °C [4,16]. POFBG was reportedly heated up to 92 °C [11] in which, so far, was 50 C [4,16]. POFBG was reportedly heated up to 92 C [11] in which, however, however, on only the grating ly th section e grating of 5 sect mm ion o was f heated 5 mm was by a he power ated b resistor y a powe to that r resist nominal or to th temperatur at nominae l temperature with unknown with un surrounding known suhumidity rroundin.g hum In that idity. In tha case the twhole case the whol POFBGe POFBG sensor sensor was not was not considered as operating at that high temperature and the POFBG sensing perfor- considered as operating at that high temperature and the POFBG sensing performance was mance w difficult to asverify difficult as t the o ver POF ify shrinkage as the PO overwhelms F shrinkage the overwhelms the contributions frcontributions om refractive index from refr change active and ind fiber ex chang thermal e anexpansion. d fiber thermal expansion. With the POFBG being pre-strained to 8000 " we first investigated the POFBG With the POFBG being pre-strained to 8000 με we first investigated the POFBG hu- humidity performance at the temperatures of 25, 35 and 45 C, respectively. At each midity performance at the temperatures of 25, 35 and 45 °C, respectively. At each temper- temperature the relative humidity of the environmental chamber was programmed to ature the relative humidity of the environmental chamber was programmed to change change with a step of 10% RH. The captured POFBG wavelength responses are shown in with a step of 10% RH. The captured POFBG wavelength responses are shown in Figure Figure 2. Figure 2. Measured POFBG humidity responses at 25, 35 and 45 °C. Figure 2. Measured POFBG humidity responses at 25, 35 and 45 C. Further experiments were carried out at higher temperature. The POFBG wavelength Further experiments were carried out at higher temperature. The POFBG wavelength was monitored with the chamber temperature set to 50 C while the relative humidity was was monitored with the chamber temperature set to 50 °C while the relative humidity was programmed to vary from 20 to 80% RH with a step increment of 20% RH. The experiment programmed to vary from 20 to 80% RH with a step increment of 20% RH. The experiment repeated at the chamber temperature of 55, 60, 65 and 70 C, respectively. The captured repeated at the chamber temperature of 55, 60, 65 and 70 °C, respectively. The captured POFBG wavelength responses are shown in Figure 3. POFBG wavelength responses are shown in Figure 3. Photonics 2021, 8, x FOR PEER REVIEW 4 of 8 Photonics 2021, 8, 180 4 of 8 Photonics 2021, 8, x FOR PEER REVIEW 4 of 8 Figure 3. Measured POFBG humidity responses at 50 °C and above. Figure 3. Measured POFBG humidity responses at 50 °C and above. Figure 3. Measured POFBG humidity responses at 50 C and above. 3. Results 3. Results 3. Results 3.1. POFBG Humidity Sensitivity 3.1. POFBG Humidity Sensitivity 3.1. POFBG Humidity Sensitivity From Figure 2 one can see that the POFBG exhibits good response against humidity From Figure 2 one can see that the POFBG exhibits good response against humidity From Figure 2 one can see that the POFBG exhibits good response against humidity change while it also shows a tendency of reduced humidity sensitivity with temperature change while it also shows a tendency of reduced humidity sensitivity with temperature change while it also shows a tendency of reduced humidity sensitivity with temperature increase. The POFBG responses in Figure 3 look noisy and do not show as clear a tendency increase. The POFBG responses in Figure 3 look noisy and do not show as clear a tendency increase. The POFBG responses in Figure 3 look noisy and do not show as clear a tendency as those in Figure 2. The noisy responses mainly arise from the strained fiber grating as those in Figure 2. The noisy responses mainly arise from the strained fiber grating as those in Figure 2. The noisy responses mainly arise from the strained fiber grating which was picking up the chamber vibration during the experiments. The vibration be- which was picking up the chamber vibration during the experiments. The vibration which was picking up the chamber vibration during the experiments. The vibration be- comes stronger when the chamber operates at higher temperature. The stabilized POFBG becomes stronger when the chamber operates at higher temperature. The stabilized POFBG comes stronger when the chamber operates at higher temperature. The stabilized POFBG wavelengths at each humidity level are summarized in Figure 4. wavelengths at each humidity level are summarized in Figure 4. wavelengths at each humidity level are summarized in Figure 4. Figure 4. Figure 4. POFBG wavelengths against humidity POFBG wavelengths against humidity change change at at dif different temperatures. ferent temperatures. Figure 4. POFBG wavelengths against humidity change at different temperatures. One can see that the first 3 sets of data show very good linear relationship between the One can see that the first 3 sets of data show very good linear relationship between One can see that the first 3 sets of data show very good linear relationship between POFBG wavelength and humidity change, giving a humidity sensitivity of 31 pm/% RH at the POFBG wavelength and humidity change, giving a humidity sensitivity of 31 pm/% the POFBG wavelength and humidity change, giving a humidity sensitivity of 31 pm/% 25 C, 23 pm/% RH at 35 C and 17 pm/%RH at 45 C, respectively. These sensitivities RH at 25 °C, 23 pm/% RH at 35 °C and 17 pm/%RH at 45 °C, respectively. These sensitiv- RH at 25 °C, 23 pm/% RH at 35 °C and 17 pm/%RH at 45 °C, respectively. These sensitiv- agree with those reported in [4]. The POFBG response at 50 C shows a greatly reduced ities agree with those reported in [4]. The POFBG response at 50 °C shows a greatly re- ities sensitivity agree with those reported in [4]. T of 8 pm/% RH. he POFBG response at 50 °C shows a greatly re- duced sensitivity of 8 pm/% RH. duced Fr sensitivity of om 55 C on8 pm/ the POFBG % RH. responses show clear nonlinearity against humidity change. From 55 °C on the POFBG responses show clear nonlinearity against humidity At 55 From C the 55 °C POFBG on the POF response B reach G response the maxima s show clear at around n 50% online RH. arAt ity higher agains temperatur t humidity es change. At 55 °C the POFBG response reach the maxima at around 50% RH. At higher change the POFBG . At 55 humidity °C the POF sensitivity BG respturns onse reach t negative he m at the axim relative a at aro humidity und 50% RH even. lower At higher than temperatures the POFBG humidity sensitivity turns negative at the relative humidity even tempera 50% RH. tures This the POFBG humi feature of POFBG dity sensiti sensor has vity tu never rns nega been observed tive at the rel befor ati e. ve humidity even lower than 50% RH. This feature of POFBG sensor has never been observed before. lower than 50% RH. This feature of POFBG sensor has never been observed before. 3.2. POFBG Temperature Sensitivity 3.2. POFBG Temperature Sensitivity The POFBG responses are summarized to show the temperature sensing performance 3.2. POFBG Temperature Sensitivity The POFBG responses are summarized to show the temperature sensing perfor- at different relative humidities, as shown in Figure 5. At a temperature below 50 C The POFBG responses are summarized to show the temperature sensing perfor- mance at different relative humidities, as shown in Figure 5. At a temperature below 50 the POFBF sensor exhibits good linear response against temperature at different rela- mance at different relative humidities, as shown in Figure 5. At a temperature below 50 °C the POFBF sensor exhibits good linear response against temperature at different rela- tive humidities. In this linear region the POFBG temperature sensitivity is calculated as °C the POFBF sensor exhibits good linear response against temperature at different rela- tive humidities. In this linear region the POFBG temperature sensitivity is calculated as 104 pm/ C at 40% RH, 116 pm/ C at 60% RH, and 133 pm/ C at 80% RH. These tive humidities. In this linear region the POFBG temperature sensitivity is calculated as Photonics 2021, 8, x FOR PEER REVIEW 5 of 8 Photonics 2021, 8, 180 5 of 8 −104 pm/°C at 40% RH, −116 pm/°C at 60% RH, and −133 pm/°C at 80% RH. These sensi- tivities are roughly two-thirds of the calculated values based on the thermo-optic coeffi- sensitivities are roughly two-thirds of the calculated values based on the thermo-optic cients of PMMA at the same relative humidity, due to the restricted expansion in the fiber coefficients of PMMA at the same relative humidity, due to the restricted expansion in the length direction, and in agreement with those reported in [14]. It can be seen that for tem- fiber length direction, and in agreement with those reported in [14]. It can be seen that for perature sensing the POFBG exhibits strong nonlinearity at temperatures above 50 °C with temperature sensing the POFBG exhibits strong nonlinearity at temperatures above 50 C with reduced reduced magn magnitude itude of tem ofptemperatur erature sene sitivity. sensitivity. Figure Figure 5. 5. POFBG POFBG wavelengths against tempe wavelengths against temperatur rature ch e change ange at diff at differ erent humidity le ent humidity levels. vels. 3.3. Analysis of POFBG Sensing Performance 3.3. Analysis of POFBG Sensing Performance As aforementioned, in this work a pre-strain of 8000 " was applied to the POFBG As aforementioned, in this work a pre-strain of 8000 με was applied to the POFBG sensor to overcome inconsistent POFBG sensor performance induced by residual drawing sensor to overcome inconsistent POFBG sensor performance induced by residual drawing stress. This pre-strain is supposed to be larger than any temperature/humidity induced stress. This pre-strain is supposed to be larger than any temperature/humidity induced fiber expansion so only refractive index change induced by temperature/humidity makes fiber expansion so only refractive index change induced by temperature/humidity makes a contribution to the POFBG wavelength change. The POFBG was pre-strained at the room a contribution to the POFBG wavelength change. The POFBG was pre-strained at the condition of ~50% RH and 25 C. The thermal expansion and the humidity induced length room condition of ~50% RH and 25 °C. The thermal expansion and the humidity induced change of POFBG can be estimated based on the property of bulk PMMA. The TEC of bulk length change of POFBG can be estimated based on the property of bulk PMMA. The TEC PMMA is the function of both temperature and humidity and is estimated as 70 10 / C of bulk PMMA is the function of both temperature and humidity and is estimated as 70 × at dry condition, 25 C to 115 10 / C at 90%RH, 70 C [17,18]. The humidity induced −6 −6 10 /°C at dry condition, 25 °C to 115 × 10 /°C at 90%RH, 70 °C [17,18]. The humidity length change is temperature independent [19] and the change rate estimated as 0.114% induced length change is temperature independent [19] and the change rate estimated as at 50%RH to 0.236% at 80%RH. This ensures the temperature/humidity induced fiber 0.114% at 50%RH to 0.236% at 80%RH. This ensures the temperature/humidity induced expansion does not exceed the pre-strain. On this condition one may analyze the POFBG fiber expansion does not exceed the pre-strain. On this condition one may analyze the performance by looking into the refractive index change of PMMA against temperature POFBG performance by looking into the refractive index change of PMMA against tem- and humidity. It should be noted that the detailed composition of dopant and copolymer perature and humidity. It should be noted that the detailed composition of dopant and used in a specific PMMA-based optical fiber is often not known. POFBG performance can copolymer used in a specific PMMA-based optical fiber is often not known. POFBG per- be approximated by that of pure PMMA, as the amount of dopant is very small. formance can be approximated by that of pure PMMA, as the amount of dopant is very The refractive index of a substance can be defined by Lorentz–Lorenz equation. It small. allows the refractive index to be obtained on the molar refraction which is often represented The refractive index of a substance can be defined by Lorentz–Lorenz equation. It as the sum of the refractions of certain constituents [20]. The refractive index n of a polymer allows the refractive index to be obtained on the molar refraction which is often repre- with moisture concentration C is expressed as [21], sented as the sum of the refractions of certain constituents [20]. The refractive index n of a polymer with moisture concentration Cm is expressed as [21], n 1 f = k r + 1 k C (2) p p m m n + 2 f −1 = +1− (2) where f is the fraction of the absorbed moisture that contributes to an increase in polymer volume, f is the critical value associated with the properties of moisture and polymer, where f is the c fraction of the absorbed moisture that contributes to an increase in polymer C = S0 H, the moisture mass in the unit volume of the polymer including moisture, S is volume, fc is the critical value associated with the properties of moisture and polymer, Cm the = S٠ moistur H, the m e solubility oisture ma of ssthe in tpolymer; he unit vorlume and k of t (ih =e p p, o m lymer , representing including mo polymer istu,rmoistur e, S is th e) e i i are the density and the specific refraction which is the molar refraction divided by the moisture solubility of the polymer; ρi and ki (i = p, m, representing polymer, moisture) are molecular the density and the specific refr weight, respectively. action which is the molar refraction divided by the molec- According to [22], r , the polymer density, is a function of moisture absorbed by ular weight, respectively. PMMA, proportional to relative humidity H. For bulk PMMA, f increases with temper- ature and f is constant over temperature (f = f at the critical temperature of 50 C) [21]. c c Specific refraction, k , generally is calculated from the measurement of refractive index [23], Photonics 2021, 8, 180 6 of 8 which implies that k is a function of temperature and humidity. In reference [21] k is i i considered constant by introducing the factor (1 f /f ) in order to simplify the mathe- kT matical processing. The term S weakly depends on temperature in a form of e where k is Boltzmann’s constant [24]. We will use this simplification to qualitatively analyze the POFBG performance. The properties of POFBG that appeared in (2) are summarized as, r , the polymer density, is a linear function of moisture; f (0 f 1) increases with temperature, and f = f at the critical temperature of 50 C; k , is a weak function of temperature and humidity, approximately constant; and S slowly decreases with temperature. At a constant temperature, the humidity dependence of the refractive index can be derived by differentiating both sides of Equation (2) as a function of H, dn (n + 2) f = 1 k S + k r (3) m p d H 6n f There are two terms on the right side of (3). For the temperature below 50 C, f < f , both terms are positive and almost constant. The POFBG refractive index is proportional to the humidity. The POFBG humidity responses therefore show good linearity. As shown in Figure 4, from 25 to 45 C the POFBG responses show good linearity and the humidity sensitivity decreases with temperature. The first term makes less contribution to the POFBG response at increased temper- ature because S decreases slowly with temperature and (1 f /f ) approaches zero with temperature increasing to 50 C. This leads to decreased humidity sensitivity. Experimental results show that at 50 C the humidity sensitivity is small. At a temperature above 50 C, f > f , the first term turns negative, further reducing the sensitivity. With temperature and/or humidity increasing, the first term may cancel the second term and the sensitivity turns negative. In Figure 4, at a temperature 50 C the humidity sensitivity starts small but positive and turns negative with increasing humidity. The dependence of k on tem- perature and humidity becomes more significant at higher temperature as the humidity sensitivity becomes smaller. As a result, the humidity sensitivity at higher temperature exhibits increasing nonlinearity. To determine the thermo-optic coefficient at a constant RH%, it can be derived by differentiating it as a function of T, dn (n + 2) f k S H 0 0 = 1 k HS f (4) dT 6n f f c c 0 0 Note here that both S and f are functions of temperature, and S and f are the corresponding derivatives. S slowly decreases with temperature, producing a small, negative S and f increases with temperature. For a temperature below 50 C, on the right side of (4) both the first term and the second are negative but the second term dominates. Therefore, in Figure 5, for the temperature below 50 C the temperature responses of the POFBG all exhibit good linearity and the magnitude of temperature sensitivity at higher humidity level is larger. The first term turns positive as f > f after 50 C and cancels part of the contribution from the second term. It leads to the reduced magnitude of temperature sensitivity. The experimental results in Figure 5 show smaller magnitude of temperature sensitivity and more nonlinearity after 50 C as the contribution of k becomes more significant. In Figure 5 the curves representing the POFBG temperature responses at different humidity levels cross at different temperatures above 50 C. This is slightly different from the case for bulk PMMA [21], in which the temperature responses of PMMA refractive index cross at 50 C, in coincidence with the critical temperature. Since the detailed figures Photonics 2021, 8, 180 7 of 8 of some parameters in (3) and (4) are not available, it is difficult to determine if this disagreement is real or distorted due to the experimental error. There also exists some disagreement between the POFBG temperature sensing per- formance and the temperature dependence of the bulk PMMA refractive index in [21]: the magnitude of the measured thermo-optic coefficient of bulk PMMA increases above the critical temperature; in contrast, the POFBG responses exhibit reduced magnitude of sensitivity above the critical temperature. The similar responses to the bulk PMMA were observed in POFBGs [25], which were verified as caused by residual drawing stress. Residual drawing stress exists in both bulk PMMA and POF [10,12]. By using pre-strained POFBG one can eliminate the effect of residual drawing stress in the POFBG wavelength response. This may indicate that the effect of residual drawing stress should be considered when looking into the thermo-optic coefficient in bulk PMMA. 4. Conclusions The POFBG sensing performances have been investigated in the extended temperature range. At below a critical temperature of ~50 C the POFBG sensor shows good linear responses and considerable sensitivities. Above the critical temperature both POFBG humidity and temperature responses exhibit nonlinearity. The POFBG humidity sensitivity could turn from positive to negative above the critical temperature. A simplified theoretical model based on Lorentz–Lorenz equation was used to qualitatively analyze the POFBG sensing responses and shows good agreement with the experimental results. When the POFBG sensor operates below the critical temperature both humidity and temperature responses show good linearity. The general expression of POFBG responses can be obtained in order to facilitate the POFBG sensing applications. However, the POFBG sensor exhibits negative humidity sensitivity above the critical temperature. This means that ambiguity could be introduced when POFBG is used for humidity sensing above the critical temperature. This ambiguity would restrict the POFBG humidity sensor from certain applications where the environment temperature is high. On other hand, the POFBG humidity negative sensitivity may be introduced in some special applications. Author Contributions: Conceptualization, W.Z. and D.J.W.; methodology, W.Z.; formal analysis, W.Z.; investigation, W.Z.; writing—original draft preparation, W.Z.; writing—review and editing, W.Z. and D.J.W.; project administration, D.J.W. Both authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. References 1. Polishuk, P. Plastic optical fibers branch out. IEEE Commun. Mag. 2006, 44, 140–148. [CrossRef] 2. Zhang, C.; Zhang, W.; Webb, D.J.; Peng, G.-D. 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