The paper is dedicated to a theoretical and experimental study of an all-glass fiber-optic microthermometer based on a Fabry-Perot interferometer. The sensing structure composed of three layers (optical fiber, borosilicate glass, and environment) is fabricated by applying borosilicate glass to the cleaved end face of a standard telecom single-mode optical fiber using a microforge. The results of experimental studies revealed that the spectral sensitivity of the first fabricated sample of the interferometer was 42 pm/°C, while the sensitivity of the second sample was 37 pm/°C. A one-dimensional mathematical model for the propagation of a plane wave through a layered structure is presented and verified, which makes it possible to calculate the spectral response of a Fabry-Perot interferometer with arbitrary structure parameters. In addition, the proposed microthermometer exhibits excellent thermal stability and repeatability, ensuring accurate temperature measurements in various environments.

1. Introduction

High-precision temperature measurement is an important task in various fields of science and technology, particularly in biomedical applications, chemical and food industries, environmental monitoring, and other domains. Fiber-optic temperature sensors offer several advantages over electronic sensors, such as compact size and weight, corrosion resistance, the absence of the need for power supply at the sensing element, immunity to electromagnetic interference, the ability to multiplex sensors and remote placement of recording equipment.

Fiber-optic temperature sensors are currently typically constructed based on Fiber Bragg gratings

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The paper introduces a Fabry-Perot interferometer with enhanced temperature sensitivity, fabricated by depositing borosilicate glass onto the end face of a standard telecommunications single-mode optical fiber using a microforge. The first section of the article presents a mathematical model of the Fabry-Perot interferometer, followed by the second section outlining the fabrication method of the microthermometer based on the optical fiber FPI. The third section presents the results of experimental investigations, and the concluding section provides the conclusions drawn from the study.

2. Mathematical Model of the Fabry-Perot Interferometer

In a general sense, the Fabry-Perot interferometer consists of two coaxial, parallel, and opposing mirrors between which a resonant standing optical wave is formed. In this case, the mirrors are formed by the boundaries between two optically transparent media. The Fabry-Perot model

[11][12]

By controlling the reflection spectrum and the positions of the maxima of the Fabry-Perot interferometer, it is possible to determine changes in the refractive index (both dielectric and magnetic permeabilities) of the layers constituting the FPI, as well as variations in the interferometer length due to external influences.

The length of a Fiber optic FPI can be comparable to the wavelength of the radiation or much larger than it. The diameter of the optical fiber does not exceed a few tens of wavelengths, and light propagation occurs along the Fiber core. This justifies the application of a one-dimensional mathematical model for the propagation of a plane wave through a layered structure (FPI). The model of such a structure is presented in Figure 1.

One-dimensional model of the Fabry-Perot interferometer

[11][12]

The wave number for each medium is defined as:

`[LATEX_FORMULA]$\kappa_i=\omega \cdot \sqrt{\varepsilon_0 \varepsilon_i \mu_0 \mu_i}=\frac{\mathrm{c}}{\lambda} \cdot \sqrt{\varepsilon_0 \varepsilon_i \mu_0 \mu_i}=\frac{\sqrt{\varepsilon_i \mu_i}}{\lambda},\left(\mathrm{c} \cdot \sqrt{\varepsilon_0 \mu_0} \equiv 1\right)$[/LATEX_FORMULA]`

where, ω – frequency, ε

`[LATEX_FORMULA]$\begin{aligned} & E_i(z)=t_i \cdot \mathrm{e}^{-j \mathrm{\kappa}_i z}+r_i \cdot \mathrm{e}^{j \mathrm{\kappa}_i z}, \\ & H_i(z)=\frac{1}{w_i}\left(t_i \cdot \mathrm{e}^{-j \mathrm{\kappa}_i z}-r_i \cdot \mathrm{e}^{j \mathrm{\kappa}_i z}\right)\end{aligned}$[/LATEX_FORMULA]`

where

The continuity of the propagation of electric and magnetic waves is characterized by the equality of these fields in each of the media at their boundaries. At the first interface, when

`[LATEX_FORMULA]$\left\{\begin{array}{l}E_1(0)=E_2(0) \\ H_1(0)=H_2(0)\end{array} \Rightarrow\left\{\begin{array}{l}t_1+r_1=t_2+r_2 \\ \frac{t_1-r_1}{w_1}=\frac{t_2-r_2}{w_2}\end{array}\right.\right.$[/LATEX_FORMULA]`

and at the second boundary, at z = h:

`[LATEX_FORMULA]$\left\{\begin{array}{l}E_2(h)=E_3(h) \\ H_2(h)=H_3(h)\end{array} \Rightarrow\left\{\begin{array}{l}t_2 \cdot \mathrm{e}^{-j \kappa_2 h}+r_2 \cdot \mathrm{e}^{j \kappa_2 h}=t_3 \cdot \mathrm{e}^{-j \kappa_3 h}+r_3 \cdot \mathrm{e}^{j \kappa_3 h} \\ \frac{t_2 \cdot \mathrm{e}^{-j \kappa_2 h}-r_2 \cdot \mathrm{e}^{j \kappa_2 h}}{w_2}=\frac{t_3 \cdot \mathrm{e}^{-j \kappa_3 h}-r_3 \cdot \mathrm{e}^{j \kappa_3 h}}{w_3}\end{array}\right.\right.$[/LATEX_FORMULA]`

Equations (3) and (4) provide with four equations for six unknown coefficients. This system of equations must be complemented. Let all the radiation that has reached the boundary at

`[LATEX_FORMULA]$t_1=1, r_3=0$[/LATEX_FORMULA]`

The complete system of four linear equations with respect to four unknown variables

`[LATEX_FORMULA]$\left[\begin{array}{cccc}1 & -1 & -1 & 0 \\ -w_2 & -w_1 & w_1 & 0 \\ 0 & \mathrm{e}^{-j \kappa_2 h} & \mathrm{e}^{j \kappa_2 h} & -\mathrm{e}^{-j \kappa_3 h} \\ 0 & w_3 \mathrm{e}^{-j \kappa_2 h} & -w_3 \mathrm{e}^{j \kappa_2 h} & -w_2 \cdot \mathrm{e}^{-j \kappa_3 h}\end{array}\right] \times\left[\begin{array}{c}r_1 \\ t_2 \\ r_2 \\ t_3\end{array}\right]=\left[\begin{array}{c}-1 \\ -w_2 \\ 0 \\ 0\end{array}\right]$[/LATEX_FORMULA]`

Of interest is only

`[LATEX_FORMULA]$r_1=\left|\frac{\cos \left(2 \pi \sqrt{\varepsilon_2 \mu_2} \frac{h}{\lambda}+\frac{i}{2} \ln \left(\frac{\left.\sqrt{\frac{\mu_2}{\varepsilon_2}}\left(-\sqrt{\frac{\mu_1}{\varepsilon_1}}-\sqrt{\frac{\mu_2}{\varepsilon_2}}+\sqrt{\frac{\mu_3}{\varepsilon_3}}\right)+\sqrt{\frac{\mu_1}{\varepsilon_1}} \sqrt{\frac{\mu_3}{\varepsilon_3}}\right)}{\left.\sqrt{\frac{\mu_2}{\varepsilon_2}}\left(-\sqrt{\frac{\mu_1}{\varepsilon_1}}+\sqrt{\frac{\mu_2}{\varepsilon_2}}+\sqrt{\frac{\mu_3}{\varepsilon_3}}\right)-\sqrt{\frac{\mu_1}{\varepsilon_1}} \sqrt{\frac{\mu_3}{\varepsilon_3}}\right)}\right)\right.}{\cos \left(2 \pi \sqrt{\varepsilon_2 \mu_2} \frac{h}{\lambda}+\frac{i}{2} \ln \left(\frac{\left.\sqrt{\frac{\mu_2}{\varepsilon_2}}\left(\sqrt{\frac{\mu_1}{\varepsilon_1}}-\sqrt{\frac{\mu_2}{\varepsilon_2}}+\sqrt{\frac{\mu_3}{\varepsilon_3}}\right)-\sqrt{\frac{\mu_1}{\varepsilon_1}} \sqrt{\frac{\mu_3}{\varepsilon_3}}\right)}{\left.\sqrt{\frac{\mu_2}{\varepsilon_2}}\left(\sqrt{\frac{\mu_1}{\varepsilon_1}}+\sqrt{\frac{\mu_2}{\varepsilon_2}}+\sqrt{\frac{\mu_3}{\varepsilon_3}}\right)+\sqrt{\frac{\mu_1}{\varepsilon_1}} \sqrt{\frac{\mu_3}{\varepsilon_3}}\right)}\right)\right.}\right|$[/LATEX_FORMULA]`

Equation (7) can be supplemented with relationships describing the dependence of the interferometer length and the coefficients of magnetic and dielectric permeability on temperature. It allows modeling the reflection spectrum of the sensitive element within the desired spectral range at given temperatures.

3. Microthermometer Fabrication

Borosilicate capillaries WPI 1B15OF-4

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The micro-pipette was secured onto a microforge constructed according to the Fonbrune design

[15]

In the initial stage, a droplet of borosilicate glass was formed on the heating coil through contact with the heated filament. Subsequently, instead of the precursor, a standard telecommunications single-mode optical fiber SMF-28 was fixed in the sample holder, and a Fabry-Perot interferometer was formed on its end face using borosilicate glass. By adjusting the temperature and manipulating the molten glass with the coil, interferometers of various shapes and geometric dimensions can be formed. Thus, two samples of end-face Fabry-Perot interferometers made from borosilicate glass were fabricated, as depicted in Figure 2, differing from each other in the interferometer length.

Microphotographs of fabricated borosilicate Fabry-Perot interferometer samples: a — Sample 1; b — Sample 2

4. Experimental investigation of the microthermometer

The experimental investigation of the fabricated microthermometer samples was conducted using a drying oven model СМ 50/250-250 ШС produced by LLC "SPM Klimat", The spectral response of the samples was registered using a fiber-optic interrogator based on the Ibsen I-MON-512 spectrometer, which was developed at the RPMT Department of KNRTU-KAI

[16]

During the experiment, the spectral responses of two fabricated Fabry-Perot interferometer samples were obtained at room temperature (23°C) as well as at temperatures ranging from 40 to 110°C in 10°C increments, as shown in Figure 3. It is important to note that the unevenness in the amplitudes of the reflectance and transmittance peaks is due to the non-ideal homogeneity of the interferometer's material.

Spectral responses of the fabricated Fabry-Perot interferometer samples:a – Sample 1; b – Sample 2

Characteristics of the peak shifts of the Fabry-Perot interferometer with respect to temperature

Dependencies of the free spectral range of Fabry-Perot interferometers on temperature

To verify the mathematical model of the Fabry-Perot interferometer presented in Section 1, the simulation of the interferometers’ spectral response was simulated, with the parameters corresponding to those of the fabricated samples: interferometer length of 70 μm and 62 μm for the Sample 1 and Sample 2, respectively; interferometer refractive index of 1.4994; optical fiber refractive index of 1.4587; surrounding medium refractive index of 1; optical fiber thermo-optic coefficient of 8.6 × 10-6Missing Mark : sup K-1Missing Mark : sup; interferometer thermo-optic coefficient of 17.2 × 10-6Missing Mark : sup K-1Missing Mark : sup; interferometer thermal expansion coefficient of 6.3 × 10-6Missing Mark : sup K-1Missing Mark : sup.

The calculated spectral responses of the models for both Fabry-Perot interferometer samples made of borosilicate glass are presented in Figure 6 at an ambient temperature of 23 °C.

Spectral reflection responses of the Fabry-Perot interferometer models at a temperature of 23 °C

5. Discussion

Nowadays, technological advancements impose increasingly demanding challenges in the field of measurement systems. In this context, the development of compact, lightweight, and thin sensitive elements not only become relevant but also critically significant. Fiber-optic sensors, in particular, exhibit substantial potential for addressing metrological tasks, including temperature measurement. Sensitive elements of fiber-optic measurement systems, based on the principles of light propagation through various structured elements within the optical fiber, seemingly align with the requirements of non-invasive measurements. Nevertheless, a range of scenarios exists where the end-face of the optical fiber proves to be excessively large, and its contact with the measurement object leads to uncontrolled dissipation of heat. This is especially important in scientific experiments, where measuring the temperature of an object is an important parameter, and it is undesirable to make changes to its thermal balance in order to avoid distortion of the results. Concurrently, it is essential to acknowledge that employing a fully glass-based optical sensitive element as a temperature sensor stems from the advantages intrinsic to glass, such as its nearly complete chemical neutrality, electromagnetic immunity across a broad frequency spectrum, biological compatibility, and the elimination of the need for electrical power supply. Thus, the challenge of designing a thinner sensitive element for the optical fiber temperature sensor appears to be relevant.

One of the ways to solve the problem is the development of sensitive elements similar to the proposed one, for example, at the end of an optical fiber with a preliminarily reduced diameter so that the diameter of the sensitive element (for example, made of borosilicate glass, Figure 2) coincides with the diameter of the fiber itself. To do this, the optical fiber can be pre-treated. For example, the cladding of an optical fiber may be chemically etched beforehand. To do so, a fragment of an optical fiber is placed in hydrofluoric acid and then is cleaved off. The second possible way to reduce the diameter of the fiber is to heat it and draw it, for example, using a coupler machine or modern welding machines. The thinner end of the optical fiber has a lower heat capacity and has less impact on the measurement object. The authors intend to continue their efforts in this direction.

6. Conclusion

The proposed Fabry-Perot fiber-optic interferometer based on borosilicate glass can serve as a temperature sensor. The fabrication process of such a sensitive element involves creating a micro-pipette from a capillary using a puller, followed by melting it and applying borosilicate glass onto the end of a single-mode optical fiber. The results of experimental studies on the two fabricated interferometer samples demonstrated that both samples exhibit close temperature sensitivity of spectral responses, which is significantly higher than the sensitivity of interferometers based on quartz glass. Specifically, the sensitivity of peak shift to temperature for sample 1 was approximately 42 pm/°C, and for sample 2, it was around 37 pm/°C. A mathematical model of the interferometer based on the scattering and transfer matrix method is presented, which allows calculating spectral responses of interferometers with any predefined parameters. The simulation results are consistent with experimental data and confirm the validity of the proposed model. Among the advantages, the sensor exhibits biocompatibility, insensitivity to moisture, and electromagnetic neutrality. The drawbacks of the presented sensitive element include the complexity of ensuring repeatability of the interferometer's geometrical parameters using the existing fabrication techniques. Authors' further research will be directed towards addressing this issue.

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