1 Introduction

FBG-based sensors are quasi-distributed and commonly used in the measurement of physical quantities such as temperature and strain. Due to the cross-sensitivity of the Bragg wavelength, a failure to perform the measurement of temperature and strain sensitivity using an FBG is the main problem of FBG sensors. In the studies in the literature, a simultaneous measurement of temperature and strain was performed for cross-sensitivity discrimination by using one or more FBGs with different sensitivities designed with different fiber structures. Two FBGs was designed using SMF28 doped with 3 mol% GeO2 in the core and SM1500 doped with 20 mol% GeO2 in the core, and was calculated their temperature and strain sensitivities by the matrix method. The simultaneous measurement of temperature and strain was performed by attaching these two FBGs with the same temperature sensitivity and different strain sensitivity end-to-end (Frazao and Santos 2004). Two FBGs at Bragg wavelengths of 1549 and 1551 nm were designed using the Optigrating application, both temperature and strain was applied to a 1549 nm FBG, and a 1551 nm FBG was only exposed to temperature The simultaneous measurement of strain and temperature was performed by taking the difference between the wavelength change resulting from the simultaneous application of temperature and strain and the wavelength change resulting from only temperature application (Elgaud et al. 2016). The TFBG was manufactured with a length of 5 mm and a tilt angle of 1.5° by using a boron-germanium doped optical fiber and was performed the simultaneous measurement of temperature and strain (Chehura et al. 2007). The TFBG was manufactured the core and cladding modes of which had different strain sensitivities and had the same temperature sensitivity, and was performed the simultaneous measurement of strain and temperature (Miao et al. 2008). The temperature sensitivities of two packaged and unpackaged TFBGs were examined and the simultaneous measurement of temperature and strain was performed (Zhang et al. 2010). The TFBG with a tilt angle of 2° was manufactured, and the temperature, strain, and the refractive index of the surrounding medium was calculated using core mode, the 25th and 28th cladding modes (Cheng et al. 2018). The internal parameters of the TFBG was examined, its optimization was provided, and the simultaneous measurement of strain and temperature of the TFBG with a tilt angle of 3 was performed (Wang et al. 2015). The TFBG with a tilt angle of 2° was designed through the Optigrating application, and only temperature measurement was performed using the LP01 and LP05 modes (Gao et al. 2014). The sensor was designed capable of measuring temperature or strain that acted as a phase-shifted grating with the FBG and TFBG written on top of each other in the same section of the optical fiber (Sun et al. 2020). The new method was suggested based on the radial collocation method to calculate the reflections in FBGs with uniform, apodized, and chirped structures (Kumar and Sharma 2019).

In this study, the shifts of wavelengths, temperature and strain sensitivities that occurred at varying temperature and strain and wavelengths of a TFBG sensor with a tilt angle of 5° designed using the Optigrating 4.2.2 software were calculated. Unlike the studies carried out in the test environment, the temperature and strain coefficients of the fiber optic cable used in the TFBG design were determined at the design stage and defined in the software. Thermal expansion and thermo-optical coefficients in the literature have been analyzed. As novelty, temperature responses of TFBG according to these coefficient changes were examined. Temperature, strain, and simultaneous temperature-strain measurements were performed, respectively, with the designed TFBG.

2 Theoretical analysis of the TFBG

As is seen in Fig. 1, TFBG fringes are tilted. While the angle (θ) of this tilt is 0° in standard FBGs, it is between 0° and 90° in the TFBG (Guo et al. 2008).

Fig. 1
figure 1

Structure of the TFBG

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In Eq. (1), the grating period is formulated along the fiber axis (Gao et al. 2014).

$${\Lambda}_{\mathrm{g}}=\Lambda /\mathrm{cos}\uptheta$$
(1)

Due to the tilted structure of the grating fringes, they increase the connection of the TFBG from the forward-propagating core mode to backward-propagating cladding modes and are connected to the core modes, and the light is coupled from the core to the cladding (Chen et al. 2008; Guo et al. 2008; Lee and Erdogan 2000).

The core mode resonance (LP01) that is also available in the transmission spectra of standard FBGs and is called Bragg and the cladding resonances (LPmn) appear simultaneously in the TFBG spectrum. The core mode of the TFBG is the resonance with the largest wavelength. It propagates along the core and, therefore, does not detect the external refractive index. The wavelengths of cladding mode resonances are less than the core mode resonance. As is seen in Fig. 2, the ghost mode resonance is evident in the transmission spectrum of TFBG and is positioned just to the left of the core mode, and therefore, it has the largest wavelength of the cladding mode resonances. This ghost mode consists of several low-order cladding modes and has many characteristics of Bragg resonance (Chen et al. 2008; Guo et al. 2008).

Fig. 2
figure 2

Transmission spectrum of TFBG

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The equations related to the wavelengths of the core and ith cladding mode resonances forming the TFBG transmission spectrum are presented in Eqs. (2) and (3) (Yucel et al. 2017a; Chen et al. 2008; Miao et al. 2008).

$${\uplambda}_{\mathrm{core}}=2{\mathrm{n}}_{\mathrm{eff},\mathrm{core}}\Lambda /\mathrm{cos}\uptheta$$
(2)
$${\uplambda}_{\mathrm{clad}}^{\mathrm{i}}=\left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)\Lambda /\mathrm{cos}\uptheta$$
(3)

2.1 Temperature and strain sensitivity of TFBG sensor

The principle of detection in TFBGs is the same as that in FBGs. When the physical changes intended to be measured are applied to the TFBG, the wavelengths of the modes in the spectrum will change. The calculation of these shifts occurring in the wavelengths is the basic principle of Bragg grating-based sensors.

Due to the tilt angle, the core resonance and cladding resonances observed in the transmission spectrum have different sensitivity to temperature and strain. Thus, the parameter intended to be measured is exposed to the TFBG, different amounts of shifts occur in the wavelengths of the core and cladding modes. Thanks to this property, it is possible to perform the simultaneous measurement of temperature and strain.

The temperature sensitivity of the core and cladding modes varies depending on the thermal expansion coefficient \(\upalpha =\frac{1}{\Lambda}\frac{{\partial \Lambda }}{{\partial \hbox{T}}}\) and thermo-optic coefficients ξ\(=\frac{1}{{\mathrm{n}}_{\mathrm{eff}}}\frac{{{\partial \hbox{n}}}_{\mathrm{eff}}}{{\partial \hbox{T}}}\) of the core and cladding. While the thermal expansion coefficient leads to a change in the grating period, the thermo-optic effect leads to a change in the effective refractive index of the fiber. When the temperature applied to the TFBG is changed by ΔΤ, the amounts of shifts occurring in the wavelengths of the core and ith cladding modes are presented in Eqs. (4) and (5) (Burunkaya and Yucel 2020; Campanella et al. 2018; Chen et al. 2008; Cheng et al. 2018; Miao et al. 2008; Nandi et al. 2019; Sengupta 2015; Yucel et al. 2017b).

$$\Delta {\uplambda}_{\mathrm{core}}=2\left(\frac{\Lambda}{\mathrm{cos}\uptheta}\ \frac{\partial {\mathrm{n}}_{\mathrm{eff}}}{{\partial {\hbox{T}}}}+\kern0.5em \frac{{\mathrm{n}}_{\mathrm{eff}}}{\mathrm{cos}\uptheta}\ \frac{{\partial \Lambda }}{{\partial {\hbox{T}}}}\right){\Delta}_{\mathrm{T}}={\mathrm{K}}_{\mathrm{core},\mathrm{T}}{\Delta}_{\mathrm{T}}$$
(4)
$$\Delta {\uplambda}_{\mathrm{clad}}^{\mathrm{i}}=\left(\frac{\Lambda}{\mathrm{cos}\uptheta}\ \frac{\partial \left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}^{\mathrm{i}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)}{{\partial \hbox{T}}}+\frac{\left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}^{\mathrm{i}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)}{\mathrm{cos}\uptheta}\ \frac{{\partial \Lambda }}{{\partial \hbox{T}}}\right){\Delta}_{\mathrm{T}}={\mathrm{K}}_{\mathrm{clad},\mathrm{T}}^{\mathrm{i}}{\Delta}_{\mathrm{T}}$$
(5)

Here, Δλcore and \(\Delta {\uplambda}_{\mathrm{cladding}}^{\mathrm{i}}\) are the change in wavelength with a temperature change of the core mode resonance and ith cladding mode resonance, respectively.

The photoelastic coefficient is expressed in terms of optical-strain coefficients as in Eq. (6) (Campanella et al. 2018; Sengupta 2015; Kipriksiz and Yucel 2021; Nandi et al. 2019).

$${\mathrm{p}}_{\mathrm{e}}=\left(\frac{{\mathrm{n}}_{\mathrm{e}\mathrm{ff}}^2}{2}\right)\left[{\mathrm{p}}_{12}-\upupsilon \left({\mathrm{p}}_{11}+{\mathrm{p}}_{12}\right)\right]$$
(6)

The strain sensitivity of the core and cladding modes varies depending on the grating period, which changes with the elongation of the fiber optic, and the change in the refractive index depending on the photoelastic effect. When the strain applied to the TFBG is changed by ΔƐ, the amounts of shifts occurring in the wavelengths of the core and ith cladding modes are presented in Eqs. (7) and (8) (Campanella et al. 2018; Chen et al. 2008; Cheng et al. 2018; Miao et al. 2008).

$$\Delta {\uplambda}_{\mathrm{core}}=2\left(\frac{\Lambda}{{\mathrm{cos}}\uptheta}\ \frac{\partial {\mathrm{n}}_{\mathrm{eff}}}{{\partial \upvarepsilon }}+\frac{{\mathrm{n}}_{\mathrm{eff}}}{{\mathrm{cos}}\uptheta}\ \frac{{\partial \Lambda }}{{\partial \upvarepsilon }}\right){\Delta}_{\upvarepsilon}={\mathrm{K}}_{{\mathrm{core}},\upvarepsilon}{\Delta}_{\upvarepsilon}$$
(7)
$$\Delta {\uplambda}_{\mathrm{clad}}^{\mathrm{i}}=\left(\frac{\Lambda}{\mathrm{cos}\uptheta}\ \frac{\partial \left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}^{\mathrm{i}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)}{{\partial \upvarepsilon }}+\frac{\left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}^{\mathrm{i}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)}{\mathrm{cos}\uptheta}\ \frac{{\partial \Lambda }}{{\partial \upvarepsilon }}\right){\Delta}_{\upvarepsilon}={\mathrm{K}}_{\mathrm{clad},\upvarepsilon}^{\mathrm{i}}{\Delta}_{\upvarepsilon}$$
(8)

If Eqs. (9) and (10) are arranged by substituting them in Eqs. (7) and (8), the following Eqs. (11) and (12) are obtained, and the amounts of shifts occurring in the wavelengths of the core and ith cladding modes can also be expressed in this way (Chen et al. 2008).

$${\mathrm{p}}_{\mathrm{core}}=-\frac{1}{{\mathrm{n}}_{\mathrm{eff},\mathrm{core}}}\frac{\partial {\mathrm{n}}_{\mathrm{eff},\mathrm{core}}}{{\partial \upvarepsilon }}$$
(9)
$${\mathrm{p}}_{\mathrm{clad}}^{\mathrm{i}}=-\frac{1}{\left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}^{\mathrm{i}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)}\frac{\partial \left({\mathrm{n}}_{\mathrm{eff},\mathrm{core}}^{\mathrm{i}}+{\mathrm{n}}_{\mathrm{eff},\mathrm{clad}}^{\mathrm{i}}\right)}{{\partial \upvarepsilon}}$$
(10)
$${\Delta \uplambda}_{\mathrm{core}}={\uplambda}_{\mathrm{core}}\left(1-{\mathrm{p}}_{\mathrm{core}}\right)\ {\Delta}_{\upvarepsilon}$$
(11)
$$\Delta {\uplambda}_{\mathrm{clad}}^{\mathrm{i}}={\uplambda}_{\mathrm{clad}}^{\mathrm{i}}\left(1-{\mathrm{p}}_{\mathrm{clad}}^{\mathrm{i}}\right)\ {\Delta}_{\upvarepsilon}$$
(12)

A two-grating detection system is used to define the relation between the wavelength shifts and the measurands is presented in Eq. (13) (Zhang et al. 2008)

$$\left(\begin{array}{c}{\Delta \uplambda}_{\mathrm{core}}\\ {}\Delta {\uplambda}_{\mathrm{clad}}^{\mathrm{i}}\end{array}\right)=\left(\begin{array}{cc}{\mathrm{K}}_{\mathrm{core},\upvarepsilon}& {\mathrm{K}}_{\mathrm{core},\mathrm{T}}\\ {}{\mathrm{K}}_{\mathrm{clad},\upvarepsilon}^{\mathrm{i}}& {\mathrm{K}}_{\mathrm{clad},\mathrm{T}}^{\mathrm{i}}\end{array}\right)\ \left(\begin{array}{c}\Delta \upvarepsilon \\ {}\Delta \mathrm{T}\end{array}\right)$$
(13)

Here, Kcore, T and Kclad, T represent the response to strain and temperature for core mode and Kclad, T and Kclad, ε for cladding mode.

3 TFBG sensor design and simulation

A TFBG with a tilt angle of 5°, the core mode of which had a wavelength of 1.55 μm, was designed by the Optigrating program using the fiber optic and grating parameter values in Table 1.

Table 1 Parameter values of TFBG
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The transmission spectrum of the TFBG in Fig. 2 was drawn by selecting LPmn modes, where m = 0,1 and n = 20. In this spectrum, it was observed that the wavelength of the ghost mode was 1.547350 μm, and the wavelength of the cladding modes varied between 1.536795 and 1.546558 μm.

As is seen in Table 2, the refractive indices of LPmn modes forming the spectrum of the TFBG are different from each other. When refractive indices are written in the formula given in Eq. (3), LPmn modes appear to be positioned as in the transmission spectrum given in Fig. 2.

Table 2 Refractive indices of LPmn modes forming the TFBG transmission spectrum
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3.1 Temperature sensitivity of the TFBG

In this section, the effect of temperature on the TFBG was examined by calculating the temperature sensitivities of the core and ghost modes. The parameter values used in the sensor design are presented in Table 3. Silica fibers were taken as a reference while selecting these value of parameters, and the thermal expansion coefficient was selected the same for core and cladding layers since the thermal expansion coefficient in this type of fibers is negligibly smaller than the thermo-optic coefficient.

Table 3 TFBG-based temperature sensor parameters
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By accepting the reference temperature as 25 °C, the temperature was varied between 25 and 75 °C with 10 °C periods. The wavelength changed depending on the expansion of the grating period due to the thermal expansion coefficient with the increase in temperature and the increase in the refractive index due to the thermo-optic coefficient. Since the thermo-optic coefficients of the core and cladding were different, the temperature sensitivity of the core and ghost modes and, therefore, the linear shifts in the wavelengths differed from each other. These values are observed in Table 4. In a temperature change of 1 °C, it is observed that a shift of approximately 75 pm occurred in the wavelength of the core mode, and a shift of approximately 65 pm occurred in the wavelength of the ghost mode.

Table 4 Wavelengths and shifts of the core and ghost modes of the TFBG with temperature application
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The values of temperature that were applied to the designed TFBG, and the amounts of linear shifts occurring in the wavelengths are presented in Fig. 3.

Fig. 3
figure 3

Wavelength change versus temperature change

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The transmission spectra of the core and ghost modes are presented in Fig. 4. The fact that the thermo-optic and thermal expansion coefficients were positive caused the spectra of the modes to shift to the right.

Fig. 4
figure 4

Transmission spectra of the core and ghost modes at different temperatures

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3.2 Effect of the thermo-optic coefficient on wavelength change

In this section of the study, the effect of the thermo-optic coefficient on the change in the wavelength of the TFBG with temperature application was examined. It was studied with different values for the thermo-optic coefficient of the core layer, with the value of 8.3e−006 by keeping the thermo-optic coefficient value of the cladding layer constant. The wavelengths of the core and ghost modes at a temperature change of 10 °C and the amounts of linear shifts in the wavelengths were obtained, and these values were presented in Table 5. Along with the increase in the thermo-optic coefficient, the core and ghost modes were positioned at larger wavelengths. Thus, the shifts in the wavelengths of both modes increased with the increase in temperature.

Table 5 Change in the thermo-optic coefficient, amounts of shifts in the wavelengths of the core and ghost modes
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The transmission spectrum of the core and ghost modes for a temperature change of 10 °C of each thermo-optic coefficient value studied is presented in Fig. 5. It was observed that the distance between the core and ghost modes increased with the increase in the thermo-optic coefficient.

Fig. 5
figure 5

Effect of the thermo-optic coefficient on the wavelength shift. (a) All core and ghost modes for ξcore values, (b) core and ghost modes for zoomed ξcore values between 1547.5 and 1551.5 nm

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3.3 Effect of the thermal expansion coefficient on wavelength change

In this section of the study, the effect of the thermal expansion coefficient on the change in the wavelength of the TFBG with temperature application was examined. It was studied with different values for the thermal expansion coefficient of the core layer, with the value of 5.5e−007 by keeping the thermal expansion coefficient value of the cladding layer constant. The wavelengths of the core and ghost modes at a temperature change of 10 °C and the amounts of linear shifts in the wavelengths were obtained, and these values were presented in Table 6. As the thermal expansion coefficient of the core increased, the core and ghost modes were positioned at larger wavelengths. The transmission spectrum of the core and ghost modes for a temperature change of 10 °C of each thermal expansion coefficient value studied is presented in Fig. 6. It was observed that the distance between the core and ghost modes increased with the increase in the thermal expansion coefficient. However, this increase was not as high as in Fig. 5. When the amounts of shifts in the wavelengths of the core and ghost modes at varying temperatures of the thermo-optic and thermal expansion coefficients were compared (Tables 5 and 6), it was determined that the thermo-optic coefficient led to a more significant change in the wavelength. Therefore, the thermo-optic coefficient was the dominant factor in the measurement of temperature sensitivity.

Table 6 Change in the thermal expansion coefficient, amounts of shifts in the wavelengths of the core and ghost modes
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Fig. 6
figure 6

Effect of the thermal expansion coefficient on the wavelength shift. (a) All core and ghost modes for αcore values, (b) core and ghost modes for zoomed αcore values between 1547.2 and 1549.2 nm, (c) core and ghost modes for zoomed αcore values between 1547.4 and 1547.8 nm

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3.4 Strain sensitivity of the TFBG

The effect of strain on the TFBG was examined by calculating the strain sensitivity of the core and ghost modes. Silica fibers were taken as a reference while selecting these value of parameters. The photoelastic parameter values of the core and cladding layers of the fiber optic used in the strain sensor design were selected the same with each other, as seen in Table 7.

Table 7 TFBG-based strain sensor parameters
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In the range of 0–500 μƐ, the strain value was increased at 100 μƐ periods and applied to the TFBG. The values of strain that were applied to the designed TFBG, and the amounts of linear shifts occurring in the wavelengths are presented in Fig. 7.

Fig. 7
figure 7

Wavelength change versus strain change

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As is seen in Table 8, the amounts of shifts in the wavelengths of the core and ghost modes were equal due to the same photoelastic parameter values for the core and cladding. Therefore, the core and cladding modes had the same strain sensitivity. It was observed that a shift of approximately 1.25 pm occurred in the wavelength of the core and ghost modes at a strain change of 1 μƐ.

Table 8 Wavelengths of the core and ghost modes of the strain-applied TFBG and amounts of shifts
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In Fig. 8, there was a linear shift to the right in the transmission spectra of the core and ghost modes for each strain value applied to the TFBG.

Fig. 8
figure 8

Transmission spectra of the core and ghost modes of the strain-applied TFBG

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3.5 Simultaneous measurement of temperature and strain

The values of temperature and strain that were simultaneously applied to the designed TFBG, the wavelengths of the core and ghost modes of the TFBG with the effect of varying temperature and strain, and the amounts of linear shifts occurring in the wavelengths are presented in Table 9 and Fig. 9. Since the research was conducted by selecting the same photoelastic parameters of the core and cladding (Table 7), the strain sensitivity of the core and cladding was the same. The thermo-optic coefficients (Table 3), which are the dominant parameters for the core and cladding temperature applications, were selected differently from each other. Thus, different amounts of shifts were obtained in the wavelengths of the core and ghost modes by ensuring that the sensitivity of the temperature in the core and ghost modes was different.

Table 9 Amounts of shifts in the wavelength of the simultaneously temperature and strain applied TFBG
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Fig. 9
figure 9

Wavelength change versus simultaneous temperature-strain change

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The transmission spectra of the core and ghost modes for the values of temperature and strain that were simultaneously applied to the TFBG are presented in Fig. 10. These transmission spectra were shifted to the right due to the positive temperature and strain parameters. Since temperature and strain were applied simultaneously, the total shift caused by both temperature and strain in the spectrum was observed.

Fig. 10
figure 10

Spectra of the core and ghost modes of the TFBG with simultaneous temperature and strain application

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By reviewing the values in Tables 4, 8, and 9, the amounts of changes in the wavelength for a temperature change of 10 °C, strain change of 100 μƐ, simultaneous temperature change of 10 °C, and strain change of 100 μƐ were calculated and presented in Table 10. When the data obtained by applying only temperature and only strain for the core and ghost modes were collected separately, it was observed that the TFBG was the same with the amounts of shift in the wavelength against simultaneous temperature and strain changes.

Table 10 Wavelength shift amounts of the TFBG in varying temperature and strain applications
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$$\left(\begin{array}{c}0.000850\ \upmu \mathrm{m}\ \\ {}0.000775\ \upmu \mathrm{m}\end{array}\right)=\left(\begin{array}{cc}{\mathrm{K}}_{\mathrm{core},\mathrm{T}}& {\mathrm{K}}_{\mathrm{core},\upvarepsilon}\\ {}{\mathrm{K}}_{\mathrm{ghost},\mathrm{T}}& {\mathrm{K}}_{\mathrm{ghost},\upvarepsilon}\end{array}\right)\ \left(\begin{array}{c}10{}^{\circ}\mathrm{C}\\ {}100\ \upmu \upvarepsilon \end{array}\ \right)$$

4 Conclusions

In the study, a TFBG sensor with a tilt angle of 5° capable of discriminating the sensitivities of simultaneously applied temperature and strain was designed using the Optigrating program. The varying temperature and the varying strain were applied to the TFBG in the first and second stages, respectively, the sensitivities of the core and ghost modes were determined separately, and the discrimination of wavelength shifts caused by simultaneous application of temperature and strain was ensured. In the measurement of the temperature sensitivity of the thermo-optic coefficient, it was observed to be the dominant parameter compared to the thermal expansion coefficient. Temperature measurements were performed between 25 and 75 °C, and strain measurements were performed between 0 and 500 μƐ. The core and ghost modes exhibited different temperature sensitivities; however, their strain sensitivities were equal. The strain sensitivity was calculated to be Kcore, Ɛ = Kghost, Ɛ = 1.25 pm/μƐ for the core and ghost modes, and the temperature sensitivity was calculated to be Kcore, T = 75 pm/°C for the core mode and Kghost, T = 65 pm/°C for the ghost mode. In the process of doping the core of the fiber optic material used in the TFGB sensor production, the additive and mole amount of the additive can be selected by taking the thermal expansion and thermo-optic coefficient values used in the study. Thus, the desired parameter values in TFBG production can be determined by the designer.