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Curr. Opt. Photon. 2024; 8(6): 602-612
Published online December 25, 2024 https://doi.org/10.3807/COPP.2024.8.6.602
Copyright © Optical Society of Korea.
Nezzar Amina1 
, Guessoum Assia2, Gerard Phillipe3,4, Lecler Sylvain3,4, Demagh Nacer-Eddine1
Corresponding author: *aminanezzar@univ-setif.dz, ORCID 0009-0005-2968-7744
**ndemagh@univ-setif.dz, ORCID 0000-0003-0804-8271
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose and theoretically investigate a fast intrinsic Mach-Zehnder (MZ) sensor, designed at the extremity of a silica single-mode optical fiber with a sensitivity that can be adapted to the considered temperature range. The sensitive part consists of a reduction of the core diameter over a tiny length (microguide), wrapped in a thin Polydimethylsiloxane (PDMS) polymer layer. The interfaces between the two guides and the distal section of the microguide form the two mirrors of the MZ interferometer. The Radiation spectrum method and COMSOL Multiphysics software are used to rigorously estimate the optical response. The numerical thermo-optical model and the results are presented and discussed. The relationship between temperature and phase shift considering the effective refractive index changes and microguide length dilation are investigated. A design method to adapt the sensor to the requirements is proposed in the −20 ℃ to +180 ℃ range. The sensor sensitivity at 1550 nm is evaluated. Due to the high PDMS thermo-optic coefficient, a temperature sensitivity of 0.6%/℃ is achievable for a 70.6 μm microguide length from −20 ℃ to +80 ℃. A response time smaller than 1 ms is demonstrated for a 180 ℃ temperature step.
Keywords: Mach-Zehnder interferometer, Optical fiber sensor, Temperature measurement, Waveguide
OCIS codes: (060.2370) Fiber optics sensors; (120.3180) Interferometry; (120.3940) Metrology; (250.0250) Optoelectronics
The precise measurement of physical and chemical parameters by optical fiber (OF) sensors is critical to the development of many applications. Therefore, several types of detection systems have been invented and proposed by researchers in different domains such as marine and seawater [1], healthcare [2], civil engineering [3, 4], steel industry [5], oil and natural gas [6], aerospace [7, 8], and military [9]. These have many interesting concepts and configurations: Interferometric [such as Fabry-Perot [10–13], Mach-Zehnder (MZ) [14, 15], Michelson [16], Sagnac [17], Fizeau [18], surface plasmon resonance (SPR) [19–21], fiber Bragg gratings (FBG) [22–24] and long period fiber grating (LPFG) [25, 26].
In the last decade, OF Mach-Zehnder sensors have been gaining in importance since they have proven their great measuring capacity and stability [27]. Also, they have shown the ability to measure different types of physical parameters: Temperature [28–30], temperature and refractive index (RI) [31, 32], pressure [33], strain [34, 35], displacement [36], gas concentration [37], humidity [38], vibration [39], etc.
Owing to the unique advantages of easy fabrication, compact size, simple structure, electromagnetic compatibility, explosive environment compatibility, and low cost, great efforts have been devoted to the study of OF MZ temperature sensors. A variety of temperature structures have been presented. Mao et al. [14], proposed an in-line Mach-Zehnder interferometer for highly sensitive temperature sensing consisting of a stub of a multimode fiber (MMF) and an up-taper; A temperature sensitivity of 113.6 pm/℃ was achieved in the range of 20 ℃ to 80 ℃ . Park et al. [40] reported an ultracompact intrinsic fiber Mack-Zehnder interferometer (MZI) with a sensitivity of 44.1 pm/℃ in the range of 20–50 ℃ by incorporating a femtosecond laser-ablated micro air-cavity. Besides sensitivity, temperature range and temperature resolution, response time is also a crucial parameter for sensors to ensure optimal performance in diverse environments and applications [41]. It has been reported that FBGs and thermometers are used to sense temperature. For example, Barrera et al. [42] reported temperature sensors using regenerated fiber Bragg gratings capable of measuring up to 1,100 ℃. To protect the delicate OF, a ceramic tube and thick metal casing are employed. A comparison of packaged and unpackaged sensors reveals response and recovery times of approximately 9 s and 22 s, respectively. Zhang et al. [43] introduce an encapsulation technology and discuss a response time analysis and experimental tests in their work. Their proposal involves packaging an FBG with a metal tube as a method of encapsulation. The sensor exhibits a response time of 48.6 ms in seawater, and they demonstrated a temperature sensitivity of 27.6 pm/℃ with linearity in the temperature range from −2 ℃ to 30 ℃. Also, a response time of 16 ms in the air was achieved by Ding et al. [44] using a compact temperature sensor based on a microfiber coupler tip (MFCT).
To develop these OF sensors, considerable attention has been paid to numerical methods and software tools in different scientific fields. They have been used to treat a wide variety of engineering problems [45], especially in the design of OF sensors and waveguides, as they contribute to a complete and rapid understanding of the proposed problem behavior [46–48]. In other words, software becomes handy because with it the behavior of the whole system can be predicted and simulated, and the performance can be enhanced by choosing the appropriate initial functional parameters [49].
As we can see, most of the sensors are based on the spectral response, which requires an expensive interrogator. In our paper, we design an intrinsic intensity modulation OF MZ temperature sensor using a single mode fiber (SMF), coupled at its distal end to a transversally single-mode microguide coated with PDMS. The influence of the geometrical and dimensional characteristics and the RI’s temperature dependency are studied in 2D.
As a result of the high thermo-optic coefficient of the PDMS cladding, our sensor has high sensitivity (0.6%/℃ for −20 ℃ to +80 ℃) to temperature changes and a fast response time (<1 ms).
A single-mode OF sensor with a simple structure is designed with a relatively acceptable temperature sensitivity that may be adaptable to the required measurement temperature range, as shown in Fig. 1.
Figure 1.Proposed Mach-Zehnder sensor structure: A polydimethylsiloxane (PDMS)-coated microguide is at the distal end of a silica single mode fiber (SMF).
Polymers are widely used in the sensing industry, and they are used in the manufacture of OF temperature sensors: Polydimethylsiloxane (PDMS) [13, 50], polyethylene terephthalate (PET), parylene-C [50], NOA61, and NOA65 [12], because they are flexible, nontoxic, chemically stable, and affordable [13, 50].
We use PDMS because of the small RI difference between PDMS and silica fiber, its large thermo-optic coefficient [50], and its low absorption at 1550 nm [13]. Proposed OF sensor was modeled in 2D by a silica dielectric waveguide inspired by an SMF28 (NA = 0.12): The core/cladding diameters and RI (ncore/nclad), are 9/125 µm and 1.46/1.45506, respectively. The basic concept of the proposed intensity modulation MZ sensor in reflection is to create a phase shift between two waves: The reference coming from the first waveguide end, and the measuring one coming from the PDMS-coated microguide. The two are supposed to be single mode.
Two reflecting surfaces (here gold layers with 100-nm thickness) can be used: The first is located at the first waveguide end, and the second is situated at the microguide extremity (see Fig. 1). Other highly conductive materials apart from gold may be considered for the coating. The input light, guided by the first waveguide, is separated into two waves with different optical paths; One is reflected by the first surface, and the second is reflected at the microguide end.
The sensor resulting intensity can be represented as:
where I1 and I2 are the reflected intensities by the two waveguide ends, respectively. λ is the free space wavelength from the source.
L is the microguide length, and neff is the effective RI of the second waveguide.
Our proposed sensor design incorporates a PDMS-coated microguide. Typically, PDMS can withstand temperatures ranging from −40 ℃ to more than 150 ℃, though this varies based on the formulation and additives [51]. For this reason, the sensor is suitable for temperatures below 200 ℃ [52] compared to silica fiber sensors [53]. When the temperature changes, both the RI and microguide length vary. These variations depend on the thermo-optic and thermal expansion coefficients of the materials.
In the general case, the RI’s temperature dependency n(T) of a material can be approximated as [54]:
The microguide length’s temperature dependence can be expressed as:
where L0 is the initial microguide length at 20 ℃, and α = 0.55 × 10−6/℃ [14] and β = 6.2 × 10−11/℃2 [54] are the first- and second-order silica thermal expansion coefficients (TEC), respectively. ξ = 8.6 × 10−6/℃ [14] and β′ = 1.62 × 10−9/℃2 [54] are the first- and second-order silica thermo-optic coefficients (TOC), respectively. To better understand the main phenomena, the second-order coefficients (β and β′) will not be taken into account.
Despite the substitution of the optical cladding by the PDMS polymer, the microguide must stay a single-mode one. For a planar waveguide, the total number of modes is obtained using the relation [55]:
where V is the normalized frequency parameter, with e the waveguide diameter, and NA the numerical aperture:
Therefore, to satisfy the condition N = 1 in the microguide, the latter should have a diameter e less than 2.23 µm where ncore = 1.46 (silica) and nclad = 1.418 (PDMS cladding RI [9]). In the following, e = 2.2 µm was selected. Both the effective RIs and the normalized frequencies were calculated for each temperature, and it was found that our microguide remains single mode up to 140 ℃. When the temperature reaches 140 ℃ and beyond, the cutoff frequency evolution shows that the microguide theoretically supports two propagation modes: A fundamental odd mode and a second, even mode. However, this second even mode cannot be excited by the odd fundamental mode of the first guide due to symmetry conservation.
After the determination of the opto-geometric sensor parameters, we simulated the sensing principle in 2D by predicting the response as a function of the temperature.
We explored several 3D rigorous calculation methods, but they were always unable to describe materials with a complex RI. Due to the necessity of incorporating metallic mirrors, we chose to perform our calculation using a 2D finite element method, following common practices.
First, the radiation spectrum method (RSM) was used to determine the effective RI of the second waveguide as a function of the temperature. It is a free beam propagation method based on a modal expansion [56, 57].
To simulate the response of our MZ sensor, we choose to perform the calculation using the commercial software COMSOL multiphysics, which is based on the finite element method, and is a numerical approach to implement approximate solutions to physical problems involving differential equations, the Helmholtz equation in the frequency domain in our case. This method is very flexible and provides complex geometry or material diversity as gold layers to be studied. The method can have a potential long computing time and required memory depending on the mesh size, which must be at least five or 10 times less than the wavelength used [15]. That is why, our simulations were performed in 2D for a planar waveguide following common practices among researchers [58, 59].
An area of 20 µm around the optical axis were studied. The mode of the first waveguide is excited using an entrance port with a diameter of 10 µm, and perfectly matched layers are used as other boundary conditions. The output signal is the backscattering intensity.
In the range of temperature between 20 ℃ and 180 ℃, the PDMS RI variation as a function of temperature [Fig. 2(a)] is opposite and around two magnitudes larger than the one of the silica core [Fig. 2(b)]. The PDMS TOC and TEC are ξ = −4.66 × 10−4/℃ [13], and α = 9.6 × 10−4/℃ [50], respectively.
Figure 2.Refractive index of (a) the polydimethylsiloxane (PDMS) and (b) the silica core fiber as a function of the temperature at λ = 1550 nm.
Using the above data, the effective RI neff of the single-mode microguide was computed as a function of the temperature using the RSM and is represented in Fig. 3.
Figure 3.Effective index of the microguide as a function of temperature at λ =1550 nm.
Despite a linear variation of the RIs as a function of temperature in the core (in silica) and in the cladding (in PDMS), the effective RI of the microguide has a nonlinear variation.
The theoretical phase shift due to the microguide length L, and RIs neff change resulting from the increase in the temperature T were calculated and are depicted in Fig. 4 in the range of temperatures between 20 to 180 ℃ using the formula:
Figure 4.Relationship between temperature and phase shift |Δϕ|, considering microguide length modification (blue curve with triangular points), effective refractive index (RI) changes (red curve with circular points), and both length modification and effective RI changes (black curve with square points). λ = 1550 nm and L0 = 70 µm, in the range of temperature between 20 ℃ to 180 ℃.
The propagation length L depends only on the silica core length. The change in the effective RI due to the two materials influences the sensor phase shift. However, in Figs. 1 and 4, we see that the temperature sensitivity of the studied sensor is mostly due to the thermo-optic effect of the PDMS cladding. So, the temperature sensitivity of the proposed structure is enormously enhanced compared to the one with a silica cladding [50].
As mentioned before, this first analysis was carried out neglecting the second-order dependencies on the temperature to better understand the sensor principle. Since they act in a quadratic manner, their influence only becomes significant at very high temperatures. Using Eqs. (2) and (3) for the phase shift calculation [Eq. (6)], both with and without considering the second-order coefficients β and β′, we observed a less than 0.04% error between the phase shift results in the two cases (3.166 rad rather than 3.165 rad shifts for a temperature increase from 20° to over 180 ℃). This would have to be taken into account in a real application.
Figure 5 illustrates the phase shift cosine variation between the two reflecting waves as a function of temperature for different microguide lengths. As expected, the phase shift in the short microguides is less than that of the long microguides. For a 180 ℃ variation, it reaches π for the first time for L = 70 µm. One can infer from the result obtained that:
Figure 5.Phase shift cosine (cos|Δϕ|) variation as a function of temperature for different microguide lengths.
For a 180 ℃ temperature variation and microguide length L = 70 µm, good visibility of the interference fringes is obtained without ambiguity on the corresponding temperature.
Higher sensitivity can be obtained for longer guides, but at smaller temperature intervals. The sensitivity for a microguide L = 100 µm is twice as great as for L = 50 µm for operating intervals from 20 ℃ to 80 ℃ and from 20 ℃ to 180 ℃, respectively.
Figure 6 shows the electric field distribution of the studied sensor. By comparing Figs. 6(a) and 6(b), it can be seen that the interference contrast is greatly influenced by the reflective surfaces. The contrast is maximized with two gold layers. According to the simulation, the contrast decrease with only one gold layer is caused by light loss through the radiation modes at the first interface (between the first and the second waveguide).
Figure 6.Electric field norm distribution [V/m] of the studied sensor at λ = 1550 nm with L0 = 70 µm, with a 100 nm gold layer thickness as a reflective mirror: (a) At the microguide end face only, (b) at the interface of the first waveguide end and microguide extremity (see Fig. 1).
Scattering parameters (or S-parameters) are complex-valued, frequency-dependent matrices describing the transmission and reflection of electromagnetic waves at different ports of devices such as filters, antennas, waveguide transitions, and transmission lines. This parameter is defined in terms of the electric field. An eigenmode analysis is required to convert an electric field pattern on a port into a scalar complex number corresponding to the voltage in transmission line theory [60].
For the S11 parameter, port 1, which is the fundamental mode at the proximal end of the first fiber in our case, is considered for both the incident field E1 and the reflected one (Ec − E1). Ec (the calculated electric field on the port) comprises both the excitation and the reflected field. The S11 parameter is then derived from these calculations [60] where A1 is the proximal section of the fiber:
Figure 7 shows the relationship between the variation in microguide length and the backscattering |S11| simulated with COMSOL when one or two mirrors (gold layer) are used. Based on the curve comparison, the contrast is higher, reaching 50%, when two reflective surfaces are used, while it is smaller than 20% with a single gold mirror.
Figure 7.Back-scattering (|S11| parameter): (a) With only the second mirror with gold at the end face. (b) With two gold mirrors.
After confirming that the temperature sensitivity of the proposed sensor is mainly attributed to the PDMS thermo-optic effect, and considering that this material can be used in a temperature range extending from −40 ℃ to more than 150 ℃ [51], we decided to explore this feature to examine our sensor’s sensitivity across a broader range and study temperatures below 0 ℃.
Thus, in this section, the temperature is varied to quantitatively determine the sensitivity of the sensor from −20 ℃ to +180 ℃ at 1550 nm.
In Fig. 8, the backscattering |S11| variation as a function of temperature is calculated with COMSOL around the inflection point corresponding to the length L0 = 70.6 µm, and the backscattering parameter S11 = 0.725 (Fig. 7). From −20 to +80 ℃ the backscattering coefficient varies from 0.86 to 0.23 and from 80 to 180 ℃, the coefficients vary from 0.23 to 0.61. A maximum sensitivity around 0.6%/℃ is achievable from −20 to +80 ℃.
Figure 8.Backscattering |S11| variation as a function of temperature. L0 = 70.6 µm.
Our analysis indicates that radial expansions minimally affect the reflection coefficient S11. The length of the initial microguide, which is 70 µm, changes by 7 nm when the temperature varies between 20 and 180 ℃. This result is obtained from the blue curve in Fig. 4 by applying Eq. (6) to the phase shift at 180 ℃. In comparison, the initial radius of the microguide, which is 1.1 µm, is altered by only 0.1 nm within the same temperature range.
As mentioned earlier, COMSOL Multiphysics can be used to simulate a variety of physical phenomena [61]. Heat transfer was investigated with a 2½-dimensional COMSOL model in the proposed asymmetric structure to determine its response time. The material properties come from the COMSOL material library. The heat capacity and thermal conductivity for silica and PDMS are 703 J/(kg.K), 1,460 J/(kg.K), and 1.38 W/(m.K), 0.16 W/(m.K), respectively.
The initial temperature in the two waveguides and at the boundaries of the first waveguide (SMF) is set at an ambient temperature (20 ℃). To simulate a temperature step (immerging the fiber tip in hot oil, for example), the temperature at the boundaries of the second waveguide was set at 180 ℃.
Figure 9 displays the temperature field distribution at the three-time steps, and the temperature in the micro-waveguide as a function of time is shown in Fig. 10. According to these results, it can be seen that with a 180 ℃ temperature step (such as immersing the fiber tip in hot oil), the micro-waveguide temperature increases from ambient to 170 ℃ in 850 µs. This is for a PDMS thickness of 9 µm. This thickness can be decreased by up to a few micrometers for a faster response. For example, a 2 µm PDMS thickness provides a response time lower than 200 µs.
Figure 9.Temperature field of the proposed structure: (a), (b) Longitudinal and Transverse view after 2 µs, respectively. (c), (d) Longitudinal and Transverse view after 20 µs, respectively. (e), (f) Longitudinal and Transverse view after 200 µs, respectively. Initial temperature is 20 ℃ and step on the second waveguide boundaries is 180 ℃.
Figure 10.Temperature over time of the micro-waveguide. L0 = 70.6 µm, temperature step is 180 ℃, and polydimethylsiloxane (PDMS) thickness is 9 µm.
In addition, another factor that will positively affect the sensor’s response time and may be taken into account in the future is the increase in the thermal conductivity of the chosen polymer with temperature, as demonstrated by Al-Khudary et al. [62] in their work.
In this work, an OF MZ temperature sensor was proposed and theoretically studied using RSM and COMSOL Multiphysics. The proposed MZ sensor is based on a PDMS-coated microguide centered on a single-mode OF end. The study shows how the sensitivity can be optimized depending on the considered temperature range controlling the microguide length. The influence of different geometric dimensional parameters, possible use of gold layers, and temperature response time are analyzed and discussed.
Due to the high PMDS thermo-optic coefficient used in the microguide thin cladding, our sensor is highly sensitive to temperature changes and has a fast response time: With a 9 µm PDMS thickness, its time response to a 180 ℃ temperature step is around 500 µs. A sensitivity of 0.6%/℃ was achieved for a 70.6 µm long microguide in a temperature range from −20 to +80 ℃. Depending on the application, longer microguides can provide greater sensitivity in a smaller temperature range.
We are looking forward to the next step, to realize a prototype of the proposed sensor. Numerous advanced and precise technologies can be used to perform such a distal microguide, such as direct femtosecond laser ablation [63], selective wet etching such as femto-print technology [64], focused ion beam microfabrication [65] and photolithography on fiber [66]. These micro-machining techniques have been used to fabricate many similar fiber sensors [67–69]. We believe our structure will provide accurate results in a variety of fields and applications.
This work was supported by the General Directorate of Scientific Research and Technological Development (DGRSDT), Ministry of Higher Education and Scientific Research (MESRS), Algeria. The authors would like to express their deepest gratitude to the ICube Research Institute, University of Strasbourg, and INSA Strasbourg, France.
General Directorate of Scientific Research and Technological Development (DGRSDT); Ministry of Higher Education and Scientific Research (MESRS), Algeria.
The authors declare no conflicts of interest.
Data supporting reported results are stored in the Applied Optics Laboratory, Institute of Optics and Precision Mechanics.
Curr. Opt. Photon. 2024; 8(6): 602-612
Published online December 25, 2024 https://doi.org/10.3807/COPP.2024.8.6.602
Copyright © Optical Society of Korea.
Nezzar Amina1 , Guessoum Assia2, Gerard Phillipe3,4, Lecler Sylvain3,4, Demagh Nacer-Eddine1
1Institute of Optics and Precision Mechanics, Ferhat Abbas University Setif1, Setif 19000, Algeria
2Quantum Electronics Laboratory, Faculty of Physics USTHB, Alger 16111, Algeria
3ICube Research Institute, University of Strasbourg, Strasbourg 67000, France
4National Institute of Applied Sciences INSA Strasbourg, Strasbourg 67000, France
Correspondence to:*aminanezzar@univ-setif.dz, ORCID 0009-0005-2968-7744
**ndemagh@univ-setif.dz, ORCID 0000-0003-0804-8271
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
We propose and theoretically investigate a fast intrinsic Mach-Zehnder (MZ) sensor, designed at the extremity of a silica single-mode optical fiber with a sensitivity that can be adapted to the considered temperature range. The sensitive part consists of a reduction of the core diameter over a tiny length (microguide), wrapped in a thin Polydimethylsiloxane (PDMS) polymer layer. The interfaces between the two guides and the distal section of the microguide form the two mirrors of the MZ interferometer. The Radiation spectrum method and COMSOL Multiphysics software are used to rigorously estimate the optical response. The numerical thermo-optical model and the results are presented and discussed. The relationship between temperature and phase shift considering the effective refractive index changes and microguide length dilation are investigated. A design method to adapt the sensor to the requirements is proposed in the −20 ℃ to +180 ℃ range. The sensor sensitivity at 1550 nm is evaluated. Due to the high PDMS thermo-optic coefficient, a temperature sensitivity of 0.6%/℃ is achievable for a 70.6 μm microguide length from −20 ℃ to +80 ℃. A response time smaller than 1 ms is demonstrated for a 180 ℃ temperature step.
Keywords: Mach-Zehnder interferometer, Optical fiber sensor, Temperature measurement, Waveguide
The precise measurement of physical and chemical parameters by optical fiber (OF) sensors is critical to the development of many applications. Therefore, several types of detection systems have been invented and proposed by researchers in different domains such as marine and seawater [1], healthcare [2], civil engineering [3, 4], steel industry [5], oil and natural gas [6], aerospace [7, 8], and military [9]. These have many interesting concepts and configurations: Interferometric [such as Fabry-Perot [10–13], Mach-Zehnder (MZ) [14, 15], Michelson [16], Sagnac [17], Fizeau [18], surface plasmon resonance (SPR) [19–21], fiber Bragg gratings (FBG) [22–24] and long period fiber grating (LPFG) [25, 26].
In the last decade, OF Mach-Zehnder sensors have been gaining in importance since they have proven their great measuring capacity and stability [27]. Also, they have shown the ability to measure different types of physical parameters: Temperature [28–30], temperature and refractive index (RI) [31, 32], pressure [33], strain [34, 35], displacement [36], gas concentration [37], humidity [38], vibration [39], etc.
Owing to the unique advantages of easy fabrication, compact size, simple structure, electromagnetic compatibility, explosive environment compatibility, and low cost, great efforts have been devoted to the study of OF MZ temperature sensors. A variety of temperature structures have been presented. Mao et al. [14], proposed an in-line Mach-Zehnder interferometer for highly sensitive temperature sensing consisting of a stub of a multimode fiber (MMF) and an up-taper; A temperature sensitivity of 113.6 pm/℃ was achieved in the range of 20 ℃ to 80 ℃ . Park et al. [40] reported an ultracompact intrinsic fiber Mack-Zehnder interferometer (MZI) with a sensitivity of 44.1 pm/℃ in the range of 20–50 ℃ by incorporating a femtosecond laser-ablated micro air-cavity. Besides sensitivity, temperature range and temperature resolution, response time is also a crucial parameter for sensors to ensure optimal performance in diverse environments and applications [41]. It has been reported that FBGs and thermometers are used to sense temperature. For example, Barrera et al. [42] reported temperature sensors using regenerated fiber Bragg gratings capable of measuring up to 1,100 ℃. To protect the delicate OF, a ceramic tube and thick metal casing are employed. A comparison of packaged and unpackaged sensors reveals response and recovery times of approximately 9 s and 22 s, respectively. Zhang et al. [43] introduce an encapsulation technology and discuss a response time analysis and experimental tests in their work. Their proposal involves packaging an FBG with a metal tube as a method of encapsulation. The sensor exhibits a response time of 48.6 ms in seawater, and they demonstrated a temperature sensitivity of 27.6 pm/℃ with linearity in the temperature range from −2 ℃ to 30 ℃. Also, a response time of 16 ms in the air was achieved by Ding et al. [44] using a compact temperature sensor based on a microfiber coupler tip (MFCT).
To develop these OF sensors, considerable attention has been paid to numerical methods and software tools in different scientific fields. They have been used to treat a wide variety of engineering problems [45], especially in the design of OF sensors and waveguides, as they contribute to a complete and rapid understanding of the proposed problem behavior [46–48]. In other words, software becomes handy because with it the behavior of the whole system can be predicted and simulated, and the performance can be enhanced by choosing the appropriate initial functional parameters [49].
As we can see, most of the sensors are based on the spectral response, which requires an expensive interrogator. In our paper, we design an intrinsic intensity modulation OF MZ temperature sensor using a single mode fiber (SMF), coupled at its distal end to a transversally single-mode microguide coated with PDMS. The influence of the geometrical and dimensional characteristics and the RI’s temperature dependency are studied in 2D.
As a result of the high thermo-optic coefficient of the PDMS cladding, our sensor has high sensitivity (0.6%/℃ for −20 ℃ to +80 ℃) to temperature changes and a fast response time (<1 ms).
A single-mode OF sensor with a simple structure is designed with a relatively acceptable temperature sensitivity that may be adaptable to the required measurement temperature range, as shown in Fig. 1.
Figure 1. Proposed Mach-Zehnder sensor structure: A polydimethylsiloxane (PDMS)-coated microguide is at the distal end of a silica single mode fiber (SMF).
Polymers are widely used in the sensing industry, and they are used in the manufacture of OF temperature sensors: Polydimethylsiloxane (PDMS) [13, 50], polyethylene terephthalate (PET), parylene-C [50], NOA61, and NOA65 [12], because they are flexible, nontoxic, chemically stable, and affordable [13, 50].
We use PDMS because of the small RI difference between PDMS and silica fiber, its large thermo-optic coefficient [50], and its low absorption at 1550 nm [13]. Proposed OF sensor was modeled in 2D by a silica dielectric waveguide inspired by an SMF28 (NA = 0.12): The core/cladding diameters and RI (ncore/nclad), are 9/125 µm and 1.46/1.45506, respectively. The basic concept of the proposed intensity modulation MZ sensor in reflection is to create a phase shift between two waves: The reference coming from the first waveguide end, and the measuring one coming from the PDMS-coated microguide. The two are supposed to be single mode.
Two reflecting surfaces (here gold layers with 100-nm thickness) can be used: The first is located at the first waveguide end, and the second is situated at the microguide extremity (see Fig. 1). Other highly conductive materials apart from gold may be considered for the coating. The input light, guided by the first waveguide, is separated into two waves with different optical paths; One is reflected by the first surface, and the second is reflected at the microguide end.
The sensor resulting intensity can be represented as:
where I1 and I2 are the reflected intensities by the two waveguide ends, respectively. λ is the free space wavelength from the source.
L is the microguide length, and neff is the effective RI of the second waveguide.
Our proposed sensor design incorporates a PDMS-coated microguide. Typically, PDMS can withstand temperatures ranging from −40 ℃ to more than 150 ℃, though this varies based on the formulation and additives [51]. For this reason, the sensor is suitable for temperatures below 200 ℃ [52] compared to silica fiber sensors [53]. When the temperature changes, both the RI and microguide length vary. These variations depend on the thermo-optic and thermal expansion coefficients of the materials.
In the general case, the RI’s temperature dependency n(T) of a material can be approximated as [54]:
The microguide length’s temperature dependence can be expressed as:
where L0 is the initial microguide length at 20 ℃, and α = 0.55 × 10−6/℃ [14] and β = 6.2 × 10−11/℃2 [54] are the first- and second-order silica thermal expansion coefficients (TEC), respectively. ξ = 8.6 × 10−6/℃ [14] and β′ = 1.62 × 10−9/℃2 [54] are the first- and second-order silica thermo-optic coefficients (TOC), respectively. To better understand the main phenomena, the second-order coefficients (β and β′) will not be taken into account.
Despite the substitution of the optical cladding by the PDMS polymer, the microguide must stay a single-mode one. For a planar waveguide, the total number of modes is obtained using the relation [55]:
where V is the normalized frequency parameter, with e the waveguide diameter, and NA the numerical aperture:
Therefore, to satisfy the condition N = 1 in the microguide, the latter should have a diameter e less than 2.23 µm where ncore = 1.46 (silica) and nclad = 1.418 (PDMS cladding RI [9]). In the following, e = 2.2 µm was selected. Both the effective RIs and the normalized frequencies were calculated for each temperature, and it was found that our microguide remains single mode up to 140 ℃. When the temperature reaches 140 ℃ and beyond, the cutoff frequency evolution shows that the microguide theoretically supports two propagation modes: A fundamental odd mode and a second, even mode. However, this second even mode cannot be excited by the odd fundamental mode of the first guide due to symmetry conservation.
After the determination of the opto-geometric sensor parameters, we simulated the sensing principle in 2D by predicting the response as a function of the temperature.
We explored several 3D rigorous calculation methods, but they were always unable to describe materials with a complex RI. Due to the necessity of incorporating metallic mirrors, we chose to perform our calculation using a 2D finite element method, following common practices.
First, the radiation spectrum method (RSM) was used to determine the effective RI of the second waveguide as a function of the temperature. It is a free beam propagation method based on a modal expansion [56, 57].
To simulate the response of our MZ sensor, we choose to perform the calculation using the commercial software COMSOL multiphysics, which is based on the finite element method, and is a numerical approach to implement approximate solutions to physical problems involving differential equations, the Helmholtz equation in the frequency domain in our case. This method is very flexible and provides complex geometry or material diversity as gold layers to be studied. The method can have a potential long computing time and required memory depending on the mesh size, which must be at least five or 10 times less than the wavelength used [15]. That is why, our simulations were performed in 2D for a planar waveguide following common practices among researchers [58, 59].
An area of 20 µm around the optical axis were studied. The mode of the first waveguide is excited using an entrance port with a diameter of 10 µm, and perfectly matched layers are used as other boundary conditions. The output signal is the backscattering intensity.
In the range of temperature between 20 ℃ and 180 ℃, the PDMS RI variation as a function of temperature [Fig. 2(a)] is opposite and around two magnitudes larger than the one of the silica core [Fig. 2(b)]. The PDMS TOC and TEC are ξ = −4.66 × 10−4/℃ [13], and α = 9.6 × 10−4/℃ [50], respectively.
Figure 2. Refractive index of (a) the polydimethylsiloxane (PDMS) and (b) the silica core fiber as a function of the temperature at λ = 1550 nm.
Using the above data, the effective RI neff of the single-mode microguide was computed as a function of the temperature using the RSM and is represented in Fig. 3.
Figure 3. Effective index of the microguide as a function of temperature at λ =1550 nm.
Despite a linear variation of the RIs as a function of temperature in the core (in silica) and in the cladding (in PDMS), the effective RI of the microguide has a nonlinear variation.
The theoretical phase shift due to the microguide length L, and RIs neff change resulting from the increase in the temperature T were calculated and are depicted in Fig. 4 in the range of temperatures between 20 to 180 ℃ using the formula:
Figure 4. Relationship between temperature and phase shift |Δϕ|, considering microguide length modification (blue curve with triangular points), effective refractive index (RI) changes (red curve with circular points), and both length modification and effective RI changes (black curve with square points). λ = 1550 nm and L0 = 70 µm, in the range of temperature between 20 ℃ to 180 ℃.
The propagation length L depends only on the silica core length. The change in the effective RI due to the two materials influences the sensor phase shift. However, in Figs. 1 and 4, we see that the temperature sensitivity of the studied sensor is mostly due to the thermo-optic effect of the PDMS cladding. So, the temperature sensitivity of the proposed structure is enormously enhanced compared to the one with a silica cladding [50].
As mentioned before, this first analysis was carried out neglecting the second-order dependencies on the temperature to better understand the sensor principle. Since they act in a quadratic manner, their influence only becomes significant at very high temperatures. Using Eqs. (2) and (3) for the phase shift calculation [Eq. (6)], both with and without considering the second-order coefficients β and β′, we observed a less than 0.04% error between the phase shift results in the two cases (3.166 rad rather than 3.165 rad shifts for a temperature increase from 20° to over 180 ℃). This would have to be taken into account in a real application.
Figure 5 illustrates the phase shift cosine variation between the two reflecting waves as a function of temperature for different microguide lengths. As expected, the phase shift in the short microguides is less than that of the long microguides. For a 180 ℃ variation, it reaches π for the first time for L = 70 µm. One can infer from the result obtained that:
Figure 5. Phase shift cosine (cos|Δϕ|) variation as a function of temperature for different microguide lengths.
For a 180 ℃ temperature variation and microguide length L = 70 µm, good visibility of the interference fringes is obtained without ambiguity on the corresponding temperature.
Higher sensitivity can be obtained for longer guides, but at smaller temperature intervals. The sensitivity for a microguide L = 100 µm is twice as great as for L = 50 µm for operating intervals from 20 ℃ to 80 ℃ and from 20 ℃ to 180 ℃, respectively.
Figure 6 shows the electric field distribution of the studied sensor. By comparing Figs. 6(a) and 6(b), it can be seen that the interference contrast is greatly influenced by the reflective surfaces. The contrast is maximized with two gold layers. According to the simulation, the contrast decrease with only one gold layer is caused by light loss through the radiation modes at the first interface (between the first and the second waveguide).
Figure 6. Electric field norm distribution [V/m] of the studied sensor at λ = 1550 nm with L0 = 70 µm, with a 100 nm gold layer thickness as a reflective mirror: (a) At the microguide end face only, (b) at the interface of the first waveguide end and microguide extremity (see Fig. 1).
Scattering parameters (or S-parameters) are complex-valued, frequency-dependent matrices describing the transmission and reflection of electromagnetic waves at different ports of devices such as filters, antennas, waveguide transitions, and transmission lines. This parameter is defined in terms of the electric field. An eigenmode analysis is required to convert an electric field pattern on a port into a scalar complex number corresponding to the voltage in transmission line theory [60].
For the S11 parameter, port 1, which is the fundamental mode at the proximal end of the first fiber in our case, is considered for both the incident field E1 and the reflected one (Ec − E1). Ec (the calculated electric field on the port) comprises both the excitation and the reflected field. The S11 parameter is then derived from these calculations [60] where A1 is the proximal section of the fiber:
Figure 7 shows the relationship between the variation in microguide length and the backscattering |S11| simulated with COMSOL when one or two mirrors (gold layer) are used. Based on the curve comparison, the contrast is higher, reaching 50%, when two reflective surfaces are used, while it is smaller than 20% with a single gold mirror.
Figure 7. Back-scattering (|S11| parameter): (a) With only the second mirror with gold at the end face. (b) With two gold mirrors.
After confirming that the temperature sensitivity of the proposed sensor is mainly attributed to the PDMS thermo-optic effect, and considering that this material can be used in a temperature range extending from −40 ℃ to more than 150 ℃ [51], we decided to explore this feature to examine our sensor’s sensitivity across a broader range and study temperatures below 0 ℃.
Thus, in this section, the temperature is varied to quantitatively determine the sensitivity of the sensor from −20 ℃ to +180 ℃ at 1550 nm.
In Fig. 8, the backscattering |S11| variation as a function of temperature is calculated with COMSOL around the inflection point corresponding to the length L0 = 70.6 µm, and the backscattering parameter S11 = 0.725 (Fig. 7). From −20 to +80 ℃ the backscattering coefficient varies from 0.86 to 0.23 and from 80 to 180 ℃, the coefficients vary from 0.23 to 0.61. A maximum sensitivity around 0.6%/℃ is achievable from −20 to +80 ℃.
Figure 8. Backscattering |S11| variation as a function of temperature. L0 = 70.6 µm.
Our analysis indicates that radial expansions minimally affect the reflection coefficient S11. The length of the initial microguide, which is 70 µm, changes by 7 nm when the temperature varies between 20 and 180 ℃. This result is obtained from the blue curve in Fig. 4 by applying Eq. (6) to the phase shift at 180 ℃. In comparison, the initial radius of the microguide, which is 1.1 µm, is altered by only 0.1 nm within the same temperature range.
As mentioned earlier, COMSOL Multiphysics can be used to simulate a variety of physical phenomena [61]. Heat transfer was investigated with a 2½-dimensional COMSOL model in the proposed asymmetric structure to determine its response time. The material properties come from the COMSOL material library. The heat capacity and thermal conductivity for silica and PDMS are 703 J/(kg.K), 1,460 J/(kg.K), and 1.38 W/(m.K), 0.16 W/(m.K), respectively.
The initial temperature in the two waveguides and at the boundaries of the first waveguide (SMF) is set at an ambient temperature (20 ℃). To simulate a temperature step (immerging the fiber tip in hot oil, for example), the temperature at the boundaries of the second waveguide was set at 180 ℃.
Figure 9 displays the temperature field distribution at the three-time steps, and the temperature in the micro-waveguide as a function of time is shown in Fig. 10. According to these results, it can be seen that with a 180 ℃ temperature step (such as immersing the fiber tip in hot oil), the micro-waveguide temperature increases from ambient to 170 ℃ in 850 µs. This is for a PDMS thickness of 9 µm. This thickness can be decreased by up to a few micrometers for a faster response. For example, a 2 µm PDMS thickness provides a response time lower than 200 µs.
Figure 9. Temperature field of the proposed structure: (a), (b) Longitudinal and Transverse view after 2 µs, respectively. (c), (d) Longitudinal and Transverse view after 20 µs, respectively. (e), (f) Longitudinal and Transverse view after 200 µs, respectively. Initial temperature is 20 ℃ and step on the second waveguide boundaries is 180 ℃.
Figure 10. Temperature over time of the micro-waveguide. L0 = 70.6 µm, temperature step is 180 ℃, and polydimethylsiloxane (PDMS) thickness is 9 µm.
In addition, another factor that will positively affect the sensor’s response time and may be taken into account in the future is the increase in the thermal conductivity of the chosen polymer with temperature, as demonstrated by Al-Khudary et al. [62] in their work.
In this work, an OF MZ temperature sensor was proposed and theoretically studied using RSM and COMSOL Multiphysics. The proposed MZ sensor is based on a PDMS-coated microguide centered on a single-mode OF end. The study shows how the sensitivity can be optimized depending on the considered temperature range controlling the microguide length. The influence of different geometric dimensional parameters, possible use of gold layers, and temperature response time are analyzed and discussed.
Due to the high PMDS thermo-optic coefficient used in the microguide thin cladding, our sensor is highly sensitive to temperature changes and has a fast response time: With a 9 µm PDMS thickness, its time response to a 180 ℃ temperature step is around 500 µs. A sensitivity of 0.6%/℃ was achieved for a 70.6 µm long microguide in a temperature range from −20 to +80 ℃. Depending on the application, longer microguides can provide greater sensitivity in a smaller temperature range.
We are looking forward to the next step, to realize a prototype of the proposed sensor. Numerous advanced and precise technologies can be used to perform such a distal microguide, such as direct femtosecond laser ablation [63], selective wet etching such as femto-print technology [64], focused ion beam microfabrication [65] and photolithography on fiber [66]. These micro-machining techniques have been used to fabricate many similar fiber sensors [67–69]. We believe our structure will provide accurate results in a variety of fields and applications.
This work was supported by the General Directorate of Scientific Research and Technological Development (DGRSDT), Ministry of Higher Education and Scientific Research (MESRS), Algeria. The authors would like to express their deepest gratitude to the ICube Research Institute, University of Strasbourg, and INSA Strasbourg, France.
General Directorate of Scientific Research and Technological Development (DGRSDT); Ministry of Higher Education and Scientific Research (MESRS), Algeria.
The authors declare no conflicts of interest.
Data supporting reported results are stored in the Applied Optics Laboratory, Institute of Optics and Precision Mechanics.
