1 Introduction

The principle sensing, in many optical waveguide based sensors, is the interaction of the medium surrounding the waveguide with the evanescent field of the propagated wave. Therefore, some of the properties of this latter (wavelength, amplitude or polarization) change and by monitoring these changes, the analyte of the concerned medium can be detected (Uniyal et al. 2023). In fact, different techniques for implementing optical waveguide thin films have been established, that involve chemical, physical, and refractive index modification methods. The simplest and most effective method to be used is the sol–gel method and dip coating technique (Butt et al. 2022a). In such technique, the refractive indices and thickness of the waveguiding film can be controlled by adjusting the molar ratio of the components and the annealing temperature, in addition to the self-stopping precursors and solvents (Butt et al. 2022b). Moreover, optical waveguide based sensors can be made from various materials depending on the purposes they serve. Recently, plasmonic refractive index sensors and nanosensors have attracted significant research interest. Therefore, a concentric double-ring resonator based MIM (Metal–Insulator–Metal) waveguides, was presented for glucose sensing capability (Farhad et al. 2020). A MIM refractive index sensor based on concentric triple ring resonator, was proposed for cancer biomarker and chemical concentration detection (Infiter et al. 2022). A cog-shaped MIM refractive index sensor studded with gold nanorods was numerically studied for temperature sensing of multiple analytes (Kazi et al. 2021a). Furthermore, (Kazi et al. 2021b) have presented a MIM nanosensors based on three circular rings for Gas-sensing and label-free detection of biomaterials. (Infiter et al. 2021a) have proposed (MIM) pressure sensor loaded with arrays of silver nanorods. In addition to MIM refractive index sensor, alternative materials based refractive index sensors have been proposed, such as, Titanium Nitride based refractive sensor (Infiter et al. 2021b), and dielectric-metal-dielectric (DMD) waveguides based nanosensors (Farhad et al. 2017).

As plasmonic refractive index sensors, optical sensors based on dielectric waveguides are well suited for chemical, gas, environmental and biological sensing. In particularly, the use of optical sensors in harsh environments is paramount. For example, high temperature gas sensors are mainly designed to solve gas detection and monitoring problems with a high operating temperature environment, such as gas turbine, nuclear power plants and automobile internal combustion engine emission (Ghosh et al. 2019; Dhall et al. 2021; Karker et al. 2017). In reality, metal oxide-based gas sensors such as SnO2, WO3, ZnO, NiO or CuO operate mostly at moderate temperatures. However, those based on Titanium dioxide (TiO2) are capable to operate above 500 °C (Fomekong and Saruhan 2021). Titanium dioxide attracts the researcher’s attention around the world, because of its unique properties. In fact, TiO2 is a non-toxic solid inorganic substance, thermally and chemically stable, non-flammable, transparent and has a large refractive index (Parrino and Palmisano 2021).

On the other hand, the use of titanium dioxide embedded in silica matrix, make produced Silica–Titania waveguide systems, highly attractive. In fact, Silica–Titania waveguides have very low propagation losses, very good stability and operation in both visible and near-infrared wavelength ranges (Karasinski et al. 2011; Butt et al. 2021a, b; Morosanova 2012; Morosanova 2016; Chiavaioli et al. 2015 and Islam et al. 2020). Moreover, a numerical analysis of Silica–Titania based reverse rib waveguide with rounded and vertical side, has appeared similar modal characteristics at the operational wavelength 1550 nm (Butt et al. 2021a, b). These properties make Silica–Titania systems suitable for integrated photonics. For example, a 1 × 2 demultiplexer for telecommunication wavelength 1550 nm and 1310 nm was proposed on silica–titania platform. This study can further open the possibilities of designing 1 × N demultiplexing devices (Butt et al. 2023).

In addition to that, refractive indices and thickness of Silica–Titania films are function of temperature. Hence, these films are promising materials for some integrated optical devices, whose, their operation principle is based on the temperature dependence of the effective refractive index of the guided modes (Nikolaev et al. 2012a,b; Pavlov et al. 2014; Pavlov et al. 2013; Nikolaev et al. 2018 and Chekhlova et al. 2006). Mainly, Silica–Titania planar waveguides systems are highly suitable for optical sensing applications (Butt et al. 2021a, b; Morosanova 2012; Morosanova 2016; Islam et al. 2018; Tiefenthaler and Lukosz 1989).

Sensitivity is the most important parameter of a planar optical sensor. In order to obtain a high value of the sensitivity and a better stability of this one as the temperature increases, our objective revolves around identifying specific waveguide parameters that minimize the temperature-dependent changes in sensitivity for the SiO2:TiO2 planar waveguide optical sensor. The study focuses, mainly, on the fundamental guided modes. Noting that, the variation of the sensitivity as a function of the temperature follows the temperature dependence of the effective refractive index, which follows the temperature dependence of the physical and geometrical parameters of the waveguide (Nikolaev et al. 2018).

2 Theoretical modeling

2.1 Dispersion equations under temperature dependence of the waveguide parameters

Figure 1 represents a schematic illustration of the investigated planar waveguide based optical sensor under temperature variation \(\left( T \right)\). The slab waveguide is composed of SiO2:TiO2 core deposited on a substrate of AF45 glass.

Fig. 1
figure 1

Evanescent wave-based SiO2:TiO2 planar waveguide optical sensor

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The dispersion equations of the planar waveguide, for both TE and TM modes, are obtained using the appropriate Maxwell equations and bouandary conditions, with considering two factors: the temperature dependence of the refractive indices of the core and substrate, \(n_{f} \left( T \right)\), \(n_{s} \left( T \right)\), respectively, and the temperature dependence of the film thickness, \(d\left( T \right)\), as given by several references, such as (Pavlov et al. 2013):

$$k_{0} d\left( T \right)\sqrt {n_{f}^{2} \left( T \right) - N\left( T \right) ^{2} } = arctan\left( {\sqrt {\frac{{N\left( T \right) ^{2} - n_{s}^{2} \left( T \right)}}{{n_{f}^{2} \left( T \right) - N\left( T \right) ^{2} }}} } \right) + arctan\left( {\sqrt {\frac{{N\left( T \right) ^{2} - n_{c}^{2} }}{{n_{f}^{2} \left( T \right) - N\left( T \right) ^{2} }}} } \right) + m\pi$$
(1)

For TE modes and

$$k_{0} d\left( T \right) \sqrt {n_{f}^{2} \left( T \right) - N\left( T \right) ^{2} } = arctan\left( {\frac{{n_{f}^{2} \left( T \right)}}{{n_{c}^{2} }}\sqrt {\frac{{N\left( T \right) ^{2} - n_{c}^{2} }}{{n_{f}^{2} \left( T \right) - N\left( T \right) ^{2} }}} } \right) + arctan\left( {\frac{{n_{f}^{2} \left( T \right)}}{{n_{s}^{2} \left( T \right)}}\sqrt {\frac{{N\left( T \right) ^{2} - n_{s}^{2} \left( T \right)}}{{n_{f}^{2} \left( T \right) - N\left( T \right) ^{2} }}} } \right) + m\pi$$
(2)

For TM modes.

Where \(n_{c}\) is the refractive index of the cover or measurand, m is the mode order, and \(N\left( T \right)\) is the effective refractive index of the guided mode, as a function of the temperature. \(k_{0}\) is the wave number in the vacuum.

To calculate the temperature dependences of the effective refractive index of Silica–Titania slab waveguide, we will employ numerical method to solve the dispersion equations, considering the temperature dependence of the waveguide parameters. These calculated effective refractive indices will then be utilized to determine the temperature dependence of the optical sensor sensitivities.

The temperature dependences of \(n_{f} \left( T \right)\), \(n_{s} \left( T \right)\) and \(d\left( T \right)\) are given by the following empirical expressions, taken from (Nikolaev et al. 2012b) and (Sainit et al. 1994):

$$n_{f} \left( T \right) = - 2.047\; \times \;10^{{ - 7}} \; \times \;T^{2} - 2.185\; \times \;10^{{ - 5}} \; \times \;T + n_{f}$$
(3)
$$n_{s} \left( T \right) = 1.5254 - 1.23\; \times \;10^{{ - 6}} \; \times \;T$$
(4)
$$d\left( T \right) = d_{0} \; \times \;\left( {1.168\; \times \;10^{{ - 6}} \; \times \;T^{2} - 1.838\; \times \;10^{{ - 5}} \; \times \;T + 1} \right)$$
(5)

where \(n_{f}\) and \(d_{0}\) are the refractive index and thickness of the core at the temperature 0 °C, respectively.

In our study, we will examine both multimode and single mode structures. For the multimode structure, we will consider two cases: one where only the fundamental modes TE0 and TM0 propagate and another where the TM1 mode can also propagates. Therefore, conditions of these three cases need to be determined.

2.2 Multimode condition

The multimode propagation exists if the cut-off thicknesses for the mth guided TM mode satisfies the following condition, deduced from (Chin 2007):

$$\frac{{arctan\left( {\sqrt {\frac{{n_{f}^{4} n_{s}^{2} - n_{c}^{2} }}{{n_{c}^{4} n_{f}^{2} - n_{s}^{2} }}} } \right) + m\pi }}{{k_{0} \sqrt {n_{f}^{2} - n_{s}^{2} } }} \le d_{cut - off} ;\; for\; TM \; modes$$
(6)
$$\frac{{arctan\left( {\sqrt {\frac{{n_{s}^{2} - n_{c}^{2} }}{{n_{f}^{2} - n_{s}^{2} }}} } \right) + m\pi }}{{k_{0} \sqrt {n_{f}^{2} - n_{s}^{2} } }} \le d_{cut - off} ;\; for\; TE\; modes$$
(7)

The condition, for which only fundamental modes, TE0 and TM0 propagate, can be, also deduced from (Chin 2007), as:

$$\frac{{arctan\left( {\sqrt {\frac{{n_{f}^{4} n_{s}^{2} - n_{c}^{2} }}{{n_{c}^{4} n_{f}^{2} - n_{s}^{2} }}} } \right)}}{{k_{0} \sqrt {n_{f}^{2} - n_{s}^{2} } }} \le d_{cut - off} < \frac{{arctan\left( {\sqrt {\frac{{n_{s}^{2} - n_{c}^{2} }}{{n_{f}^{2} - n_{s}^{2} }}} } \right) + \pi }}{{k_{0} \sqrt {n_{f}^{2} - n_{s}^{2} } }}$$
(8)

Noting that for multi-mode propagation, the condition of the mth guided modes sufficient, because their cut-off thicknesses are always greater than those of TE modes of the same order.

2.3 Single mode condition

It is possible to propagate a single fundamental mode TE0 in the case where d satisfies the cut-off thicknesses condition, deduced from (Chin 2007):

$$\frac{{arctan\left( {\sqrt {\frac{{n_{s}^{2} - n_{c}^{2} }}{{n_{f}^{2} - n_{s}^{2} }}} } \right)}}{{k_{0} \sqrt {n_{f}^{2} - n_{s}^{2} } }} \le d_{cut - off} < \frac{{arctan\left( {\sqrt {\frac{{n_{f}^{4} n_{s}^{2} - n_{c}^{2} }}{{n_{c}^{4} n_{f}^{2} - n_{s}^{2} }}} } \right)}}{{k_{0} \sqrt {n_{f}^{2} - n_{s}^{2} } }}$$
(9)

2.4 Temperature dependence of the sensitivity of the slab waveguide based sensor

Sensitivity is the change of the sensor output signal in response to unit change in a property of the sensor (the concentration of an analyte surrounded the surface of the sensor). A parameter defines the ability of a sensor to transduce an input signal to an output one. The sensitivity is function of the substance to be detected and the kind of the waveguide transducer. The sensitivity is deduced from the expression:

$$S = \left( {\frac{{\partial n_{c} }}{\partial N\left( T \right)}} \right)^{ - 1}$$
(10)

By differentiating the characteristics Eqs. (1) and (2) and considering the last Eq. (10), we get:

$$S_{TE} = \sqrt {\frac{{1 - a_{c} \left( T \right)}}{1 - X\left( T \right)}} \frac{X\left( T \right)}{{a_{c} \left( T \right)\sqrt {a_{c} \left( T \right) - X\left( T \right)} }}\frac{1}{{\left[ {K\left( T \right) + \frac{1}{{\sqrt {a_{s} \left( T \right) - X\left( T \right)} }} + \frac{1}{{\sqrt {a_{c} \left( T \right) - X\left( T \right)} }}} \right]}}$$
(11)

For TE modes, and

$$S_{TM} = \frac{{\left[ {2\sqrt {\frac{1 - X\left( T \right)}{{1 - a_{c} \left( T \right)}}} - \sqrt {\frac{{1 - a_{c} \left( T \right)}}{1 - X\left( T \right)}} } \right]\frac{{X\left( T \right)\left( {1 - a_{c} \left( T \right)} \right)}}{{a_{c} \left( T \right)\sqrt {\left( {a_{c} \left( T \right) - X\left( T \right)} \right)} \left[ {1 - X\left( T \right)\left( {2 - a_{c} \left( T \right)} \right)} \right]}}}}{{\left[ {K^{\prime}\left( T \right) + \frac{{\left( {1 - a_{c} \left( T \right)} \right)}}{{\sqrt {\left( {a_{c} \left( T \right) - X\left( T \right)} \right)} \left[ {1 - X\left( T \right)\left( {2 - a_{c} \left( T \right)} \right)} \right]}} + \frac{{\left( {1 - a_{s} \left( T \right)} \right)}}{{\sqrt {\left( {a_{s} \left( T \right) - X\left( T \right)} \right)} \left[ {1 - X\left( T \right)\left( {2 - a_{s} \left( T \right)} \right)} \right]}}} \right]}}$$
(12)

For TM modes.

Where the following parameters are function of the temperature

$$K\left( T \right) = \frac{{arctan\sqrt {\frac{{a_{s} \left( T \right) - X\left( T \right)}}{X\left( T \right)}} + arctan\sqrt {\frac{{a_{c} \left( T \right) - X\left( T \right)}}{X\left( T \right)}} + m\pi }}{{\sqrt {X\left( T \right)} }}$$
(13)
$$K^{\prime } \left( T \right) = \frac{{arctan\frac{1}{{1 - a_{c} \left( T \right)}}\sqrt {\frac{{a_{c} \left( T \right) - X\left( T \right)}}{X\left( T \right)}} + arctan\frac{1}{{1 - a_{s} \left( T \right)}}\sqrt {\frac{{a_{s} \left( T \right) - X\left( T \right)}}{X\left( T \right)}} + m\pi }}{{\sqrt {X\left( T \right)} }}$$
(14)
$$a_{c} \left( T \right) = 1 - \frac{{n_{c}^{2} }}{{n_{f}^{2} \left( T \right)}};\;X\left( T \right) = 1 - \frac{{N^{2} \left( T \right)}}{{n_{f}^{2} \left( T \right)}}\;a_{s} \left( T \right) = 1 - \frac{{n_{s}^{2} \left( T \right)}}{{n_{f}^{2} \left( T \right)}}$$
(15)

3 Results and discussions

We have developed a Matlab program to calculate the temperature dependence of the sensitivity and the effective refractive index for both TE and TM modes in a SiO2:TiO2 planar waveguide based optical sensor. This temperature dependence follows the temperature dependence of the refractive indices of the waveguiding film and substrate, as well as the temperature dependence of the waveguiding film thickness. Our focus is on the TE0 and TM0 modes, due to their higher sensitivities compared to higher order modes. The chosen numerical values of the core refractive index \(n_{f}\) are (1.90, 2.0, and 2.10) (Butt et al. 2022b). The practical values of the cover refractive indices \(n_{c}\) were selected as (1.4394, 1.47, and 1.501); these are corresponding to the glucose solution, vegetable oil and benzene materials, respectively. Noting that other materials of the cover can be chosen if their refractive indices are lower than those of the core and substrate. The appropriate core thicknesses will be determined based on the cut-off condition of the modes in each case. The temperature range considered in our study extends from 20 to 500 °C.

Figure 2 represents the sensitivity variation of the TE0 (blue curves) and TM0 (red curves) modes as a function of the temperature for different wavelengths of the source, solid lines for the source wavelength of 1550 nm and dashed lines for the source wavelength of 632.8 nm. The Figure demonstrates that the TM0 mode exhibits higher sensitivities compared to the TE0 mode. Additionally, the sensitivities decrease as the temperature increases. It should be noted that the selected core thickness is optimized for the source wavelength of 1550 nm, resulting in higher sensitivities for this wavelength compared to the source wavelength of 632.8 nm, this is true for both TE and TM polarizations.

Fig. 2
figure 2

Variation of the sensitivities of the TM0 and TE0 modes according to the temperature for different source wavelengths (nf = 2.10, nc = 1.501, d0 = 167 nm)

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It is worth to note that during the subsequent study, we will use a source with the wavelength of 1.55 µm. This choice is made based on technical considerations, since it is generally more practical to manufacture waveguides with larger thicknesses rather than those with smaller thicknesses. Knowing that longer wavelengths require greater core thicknesses and vice versa.

3.1 Sensitivity temperature-dependence of TM modes in multimode structure

3.1.1 Random choice of waveguide parameters

In this section, we will first examine the case where only the TE0 and TM0 modes exist. We will specifically focus on the sensitivity of the TM0 mode, as it exhibits greater sensitivity compared to the TE0 mode. We aim to investigate how the sensitivities of the TM0 mode vary with temperature, considering different random, physical and geometric parameters of the waveguide. Next, we will study the case of the TM1 mode. It is important to note that the chosen core thicknesses satisfy the multimode condition specified by Eq. (6), for TM0 and TM1 modes.

Figures 3, 4 and 5 represent the variation of the sensitivities of the TM0 mode as a function of the temperature for different core refractive indices, measurand refractive indices and core thicknesses, successively. It should be noted that the chosen core thickness for each figure satisfies the mode propagation condition. In fact, Figs. 3 and 4 show an increase in the sensitivities by increasing the refractive indices of the core and substrate. Whereas, Fig. 5 demonstrates that the sensitivities decrease as the core thicknesses increase. On the other hand, the curves indicate that the increase in the temperature leads to a diminution in sensitivities, particularly at high temperatures. It should be emphasized that this decrease in the sensitivity, as the temperature increases, is more pronounced for the higher values of the waveguide parameters. The difference between the maximum and minimum value of the sensitivity reaches 0.1, in some cases. These figures confirm that the arbitrary selection of the physical and geometrical parameters of the waveguide can have a significant impact on the sensitivity of the sensors and their variation with respect to the temperature.

Fig. 3
figure 3

Variation of the sensitivities of the TM0 mode according to the temperature for different core refractive indices (λ = 1550 nm, nc = 1.501, d0 = 230 nm)

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Fig. 4
figure 4

Variation of TM0 mode sensitivities as a function of the temperature for different cover refractive indices (λ = 1550 nm, nf = 2.1, d0 = 270 nm)

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Fig. 5
figure 5

Variation of TM0 mode sensitivities as a function of the temperature for different core thicknesses (λ = 1550 nm, nf = 2.1, nc = 1.501)

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3.1.2 Optimization of the waveguide parameters

3.1.2.1 Case of TM0 mode for different core refractive indices

In order to increase the sensitivity of the sensor and to minimize the effect of the temperature, we have chosen optimal thicknesses for each refractive index of the core. These thicknesses are those where the sensitivities are maximum at the starting temperature T = 20 °C, and the thicknesses close to the cut-off thicknesses of the TM0 mode.

Through the plot of the sensitivity versus core thicknesses at the temperature of 20 °C, and for various values of core refractive indices, we sought to identify the core thicknesses (d0max) those yield maximum sensitivities. According to the Fig. 6, these are 223 nm, 210 nm and 200 nm, corresponding to the core refractive indices of 1.9 (red curve), 2.0 (blue curve) and 2.1 (black curve), respectively.

Fig. 6
figure 6

Sensitivities of the TM0 mode according to the core thicknesses for different core refractive indices at T = 20 °C (λ = 1550 nm, nc = 1.501)

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Figure 7 presents the variation of the maximum sensitivities of the TM0 mode as a function of the temperature for different values of the core refractive indices and the corresponding core thicknesses determined previously. The most prominent observation in this figure is that the sensitivities of the TM0 mode, across all three-core refractive indices, decrease in the same manner and magnitude, as the temperature increases. However, the decrease in sensitivity is more important at higher temperatures. The difference between the maximum and minimum sensitivity values is 0.035, for all three curves. Nevertheless, this difference in sensitivities is relatively smaller compared to the cases where core thicknesses were randomly chosen. It is worth noting that below 100 °C, the sensitivities remain constant, while, beyond this temperature, they progressively decrease.

Fig. 7
figure 7

Temperature dependence of the sensitivities of the TM0 mode for different core refractive indices and for d0max (λ = 1550 nm, nc = 1.501)

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To achieve an improved stability of the sensitivity across the investigated temperature range, we have plotted the sensitivities for core thicknesses close to the cut-off thicknesses (d0min) of the TM0 mode. Specifically, we considered core thicknesses of 202 nm, 182 nm and 167 nm, corresponding to the core refractive indices 1.9, 2 and 2.1, respectively. The Fig. 8 exhibits nearly stable sensitivity curves across a wide temperature range (250 °C, 350 °C and 450 °C) for the respective core thicknesses of 202 nm, 182 nm and 167 nm. Furthermore, the sensitivities decrease for high temperatures are relatively smaller than that observed in the previous Fig. 7. The differences between the maximum and minimum value of the sensitivity are 0.005, 0.015 and 0.02, corresponding to the core refractive indices 2.1, 2 and 1.9, respectively. It is worth mentioning, that the curve for the largest core refractive index is the most stable.

Fig. 8
figure 8

Temperature dependence of the sensitivities of the TM0 mode for different core refractive indices and for d0min (λ = 1550 nm, nc = 1.501)

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3.1.2.2 Case of TM0 mode for different cover refractive indices

In this section, we will determine the optimal cover refractive index that provides the highest and most stable sensitivity as well as the conditions under which it occurs. The studied cover refractive indices are 1.501, 1.47 and 1.4394, while the used core refractive index is the highest (2.1), as it gives the best sensitivity. Firstly, we will identify the optimal thicknesses that result in the maximum sensitivities for each cover refractive index. These thicknesses are defined from Fig. 9, which displays the sensitivities of the TM0 mode as a function of the thicknesses for three values of the measurand refractive indices. The deduced optimum thicknesses are 202.1 nm, 240.3 nm and 262.1 nm corresponding to the cover refractive indices of 1.501 (black curve), 1.47 (blue curve) and 1.4394 (red curve), respectively. Noting that, the highest sensitivity is achieved for the highest refractive index of the measurand, as it is closer to the substrate refractive index ns = 1.5254. This proximity of refractive indices enhances the interaction between the waveguide and the surrounding medium, leading to a higher sensitivity of the sensor (Cherouana et al. 2019).

Fig. 9
figure 9

Sensitivities of the TM0 mode as a function of the core thicknesses for different cover refractive indices at T = 20 °C (λ = 1550 nm, nf = 2.1)

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In Fig. 10, the sensitivity of the TM0 mode is presented as a function of the temperature, for different values of the measurand refractive indices, and for the core thicknesses that provide maximum sensitivities. The Figure shows that the sensitivities exhibit a similar decreasing trend when the temperature increases, but with different magnitudes. Notably, the measurand refractive index with the highest value exhibits the slowest rate of sensitivity decrease. The differences between the maximum and minimum value of the sensitivity are 0.04, 0.055 and 0.06, corresponding to the measurand refractive indices 1.501, 1.47 and 1.4394, respectively.

Fig. 10
figure 10

Temperature dependence of the sensitivities of the TM0 mode for different cover refractive indices and for d0max (λ = 1550 nm, nf = 2.1)

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On the other hand, we also examined the temperature dependence of the sensitivity at different cover refractive indices, for core thicknesses close to the cut-off thicknesses of the TM0 mode (see Fig. 11). These thicknesses are 167 nm, 170 nm and 174 nm, corresponding to the cover refraction indices of 1.501, 1.47 and 1.4394, respectively. In the case of a measurand refractive index of 1.501, the sensitivity remained relatively stable at around 0.41. However, for measurand refractive indices of 1.47 and 1.4394, the sensitivities increased with temperature. The Fig. 11 also indicates that the sensitivity increases and tends towards stabilization by increasing the measurand refractive index, provided that the latter is lower than the refractive index of the substrate.

Fig. 11
figure 11

Temperature dependence of the sensitivities of the TM0 mode for different cover refractive indices and for d0min (λ = 1550 nm, nf = 2.1)

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3.1.2.3 Case of TM1 mode for different core refractive indices

Figure 12 illustrates the change in the sensitivities as a function of the temperature for the TM1 mode, for three core refractive indices and their thicknesses corresponding to the maximum sensitivities. The figure shows that the sensitivities are maximum and almost stable at low temperatures (< 100 °C). However, they rapidly decrease as the temperature increases. Moreover, the curves indicate that, the higher core refractive indices result in a faster drop in sensitivity. The differences between the maximum and minimum sensitivities are 0.135, 0.12 and 0.105 for the core refractive indices of 2.1, 2 and 1.9, respectively. This emphasizes that the decrease in sensitivities with temperature is more pronounced for the mode TM1 compared to the TM0 mode.

Fig. 12
figure 12

Temperature dependence of the sensitivities of the TM1 mode for different core refractive indices and for d0max (λ = 1550 nm, nc = 1.501)

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In Fig. 13, the temperature dependence of the sensitivities for TM1 mode are plotted considering core thicknesses near to the cut-off thicknesses of the TM1 mode. The figure reveals that the sensitivities initially increase with temperature until reaching certain threshold values, after which they slowly decrease. It should be noted that the sensitivity values at low temperatures are very small.

Fig. 13
figure 13

Temperature dependence of the sensitivities of the TM1 mode for different core refractive indices and for d0min (λ = 1550 nm, nc = 1.501)

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3.2 Sensitivity temperature-dependence of TE0 mode in single mode structure

In this case, we will focus on the TE0 mode as the only propagating mode. We aim to investigate whether the behavior of the TE0 single mode is similar to that of the TM0 mode, regarding the variation of sensitivity with temperature for different physical and geometric parameters of the waveguide.

As in the case of TM0 mode, we have chosen optimal thicknesses for each refractive indices of the core and the cover. These thicknesses are those where the sensitivities are maximum at the starting temperature T = 20 °C, and the thicknesses close to twice the cut-off thicknesses of the TE0 mode. The choice of twice the cut-off thicknesses for the TE0 mode is a result of simulation; in fact, we have plotted S versus T for different values of thickness (d), far and close to the cut-off thicknesses of the TE0 mode. Our findings indicate that employing a thickness (d) approximately double the cut-off value yields the highest and most stable sensitivity.

3.2.1 Case of TE0 mode for different core refractive indices

Firstly, the optimal core thicknesses (d0max) those yield maximum sensitivities at T = 20 °C are deduced from Fig. 14, these are 130 nm, 100 nm and 86.72 nm, corresponding to the core refractive indices of 1.9 (red dashed curve), 2.0 (blue dashed curve) and 2.1 (black dashed curve), respectively.

Fig. 14
figure 14

Sensitivities of the TE0 mode as a function of the core thicknesses for different core refractive indices at T = 20 °C (λ = 1550 nm, nc = 1.501)

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Figure 15 depicts the sensitivity change of the TE0 mode with respect to the temperature for different core refractive indices and for the thicknesses d0max defined previously. The figure shows that the sensitivities decrease with increasing temperature but not at the same rate for all cases. Notably, the sensitivity with the core refractive index 2.0 exhibits the widest stabilization range, extending up to 200 °C. Beyond this temperature, there is a decrease of 0.0125 compared to the initial value of sensitivity. For the core refractive index of 2.1, the sensitivity stabilization persists up to 150 °C, followed by a decrease of 0.015 at 500 °C. Whereas, the difference between the maximum and minimum value of the sensitivity is almost 0.02, corresponding the core refractive index value of 1.9. It can be observed that the decrease in sensitivities of the TE0 mode with increasing temperature, are small, compared to those of the TM0 mode. This suggest that the TE0 mode is less affected by temperature variation, making it more suitable for certain applications where temperature stability is crucial.

Fig. 15
figure 15

Temperature dependence of the sensitivities of the TE0 mode for different core refractive indices and for d0max (λ = 1550 nm, nc = 1.501)

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In order to enhance sensitivities at elevated temperatures, we have presented in Fig. 16, the sensitivities corresponding to core thicknesses close to twice the cut-off thicknesses of the TE0 mode, considering various core refractive indices. The plotted curves reveal that as temperature rises, the sensitivities exhibit a notable increase, especially for higher core refractive indices.

Fig. 16
figure 16

Temperature dependence of the sensitivities of the TE0 mode for different core refractive indices and for 2*d0min (λ = 1550 nm, nc = 1.501)

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3.2.2 Case of TE0 mode for different cover refractive indices

As in the previous case, the core thicknesses (d0max) corresponding the maximum sensitivities at T = 20 °C, are deduced from Fig. 17, these are 124.5 nm, 111.3 nm and 89.96 nm, for the cover refractive indices of 1.4394 (red dashed curve), 1.47 (blue dashed curve) and 1.501 (black dashed curve), respectively.

Fig. 17
figure 17

Sensitivities of the TE0 mode as a function of the core thicknesses for different cover refractive indices at T = 20 °C (λ = 1550 nm, nf = 2.1)

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The temperature-dependent sensitivities of the mode TE0 are depicted in Fig. 18 for various refractive indices of the measurand. The core thicknesses (d0max) chosen for analysis are those determined previously. Notably, the curves appear to be identical and the sensitivity gradually decreases with increasing temperature. The difference between the maximum and minimum sensitivity is 0.02, indicating acceptable stability. Furthermore, Fig. 19 illustrates the sensitivity plot for core thicknesses near twice the cut-off thicknesses corresponding to each refractive index of the measurand. This approach yields even more favorable results, particularly in terms of sensitivity stabilization. Specifically, for a cover refractive index of 1.501, the sensitivity exhibits a slightly increase with temperature. Conversely, for a cover refractive index equals to 1.4394, the sensitivity experiences a slight decrease of approximately 0.015. Nevertheless, the sensitivity remains nearly constant at around 0.29, for a cover refractive index of 1.47.

Fig. 18
figure 18

Temperature dependence of the sensitivities of the TE0 mode for different cover refractive indices and for d0max (λ = 1550 nm, nf = 2.1)

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Fig. 19
figure 19

Temperature dependence of the sensitivities of the TE0 mode for different cover refractive indices and for 2*d0min (λ = 1550 nm, nf = 2.1)

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Table 1 summarizing the best numerical results obtained for different modes, including the highest and more stable values of the sensitivities, their optimized thicknesses and their corresponding refractive indices.

Table 1 Summarizing of the best numerical results obtained for different modes
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4 Conclusion

In summary, this study demonstrates that the random choice of physical and geometrical parameters of the waveguide could significantly decrease the sensitivity of the sensor as the temperature increases. However, the appropriate selection of these parameters, particularly, the core thickness, not only made it possible to increase the sensitivity but also to extend its range of stabilization with respect to the temperature, and in certain cases the sensitivity could even be notably enhanced with rising the temperature. The most favorable results, in terms of sensitivity value and stability are obtained, by selecting a larger core refractive index with a measurand refractive index closest to that of the substrate. Regarding the geometrical parameters, optimal results can be achieved for the core thicknesses located between the core thickness where the sensitivity is maximum at room temperature and the cut-off thickness of the TM0 mode. Similarly, for the single mode structure, the best results can be attained for core thicknesses positioned between the thickness where the sensitivity is maximum at room temperature and twice the cut-off thickness of the TE0 single mode. This study demonstrate that the SiO2:TiO2 waveguide based optical sensor can be used in high temperature environment with high and quasi-stable sensitivity, provided that the waveguide’s parameters are suitably defined. This holds true for both TE and TM polarizations.