Efficient sensings of temperature, refractive index, and distance measurement using the cubic-nonlinear optoelectronic oscillators

In this work, we use the cubic-nonlinear optoelectronic oscillator (CNOEO) in three different configurations to successfully perform the sensings of temperature, refractive index, and distance measurement with high precision. From our results, the use of the CNOEO through the variation of the values of the cubic-nonlinear parameter increases the sensitivity of the different sensings and the measurement carried out compared to the results obtained with the standard OEO which is a particular CNOEO featuring a cubic-nonlinear parameter equal to zero.


Introduction
Optoelectronic oscillators (OEOs) have inspired numerous applications and technological aims such as ultra-stable microwave generation, neuromorphic computing, signal processing, photonic integration circuits, random numbers generation, and chaos communications (see Larger 2013;Chembo et al. 2019;Hao et al. 2020;Talla Mbé et al. 2021 and references therein for a comprehensive review). Recently, the scope of technology and applications has widened to incorporate sensing, measurement, and detection (Zou et al. 2016;Yao 2017;Wu et al. 2018) owing to the fact that typically, an OEO is a high-Q device capable of producing microwave signals with ultralow phase noise and high frequency (Zhang et al. 2018). Indeed, with OEOs it is possible to carry out the measurement or 1 3 311 Page 2 of 13 sensing of length, refractive index, temperature, angular velocity, load, strain, low-power RF signal, magnetic field, amongst others (see Zhang et al. 2018Zhang et al. , 2021Liu et al. 2019;Zhu et al. 2019;Xie et al. 2019;Xu et al. 2019;Ming et al. 2021;Zhang et al. 2020;Feng et al. 2019 and references therein). For this purpose, several architectures were proposed for the efficiency of the method. For instance, the authors of Feng et al. (2019) achieved a strain-insensitive temperature sensor with a sensitivity of 1.00745 MHz/ • C based on an optoelectronic oscillator incorporating both the phase and the intensity optical modulators. Besides, Zhang et al. achieved a sensitivity of the angular velocity of 51.8 kHz/(rad/s) using an OEO incorporating a Sagnac interferometer (Zhang et al. 2018).
The fundamental principle of using OEO for optical metrology is to code the quantity to evaluate, such as temperature, strain, transverse load, refractive index, and more, to the frequency shift of oscillating signal of OEOs. Two mechanisms are possible (Feng et al. 2019): The first one consists of using the band-pass of the microwave photonic filter both as an oscillation frequency selection element as well as a sensing/measurement element. Even if it gives high sensitivity for optical metrology, it is not easy to implement experimentally. The second one known as the time-delay mechanism is quite easy since it relies on the direct change in the oscillator's loop when affected by the quantity to sense, detect, or measure.
Thus, this latter technique is the most developed one . Following this, an OEO with optimal frequency could play an important role in sensing and measurement processes. Recently, an OEO displaying dynamics predominated by high-frequency limit-cycle oscillations was proposed (Talla Mbé et al. 2019). Such dynamics were made possible by substituting the standard band-pass filter in the electrical path of the standard OEO with a cubic-nonlinear band-pass filter. The oscillator is known as the cubicnonlinear optoelectronic oscillator (CNOEO) and has not yet been used for sensing and measurement.
In this paper, we theoretically aim to increase the sensitivity of the sensor/measurerbased OEOs using the CNOEO. The distance measurement and two main sensings are investigated, namely, the temperature and the refractive index. The paper is organized as follows: In Sect. 2, the cubic-nonlinear optoelectronic oscillator (CNOEO) is presented. Applications to temperature, refractive index sensings, and the distance measurement are discussed in Sects. 3, 4, and 5, respectively. Finally, a conclusion is given in Sect. 6.

The cubic-nonlinear optoelectronic oscillator (CNOEO)
The CNOEO used for this work is displayed in Fig. 1a (Talla Mbé et al. 2019). The CNOEO is made up of a laser diode, a polarization controller (PC), a Mach-Zehnder modulator (MZM), an optical delay line, a photodiode (PD), a cubic-nonlinear band-pass filter (CNBPF), a voltage subtractor (VS), an amplifier (Amp), a microwave coupler (MC), and an oscilloscope to visualize the oscillating signal. The CNBPF is made of an inductor, a resistance, and a nonlinear capacitor (NC). The structure of the nonlinear capacitor is shown in Fig. 1b: an operational amplifier (U), two capacitors C 1,2 , one resistor r, and eight junction diodes can be identified. The functioning of the CNOEO will be specifically given for each application treated in this paper. Detailed explanations of the main CNOEO can be found in Talla Mbé et al. (2019), Kamaha et al. (2020). Nevertheless, it is important to notice that the dynamics of the CNOEO is governed by the following integro-delay differential equation (Talla Mbé et al. 2019;Kamaha et al. 2020): where the parameters are expressed in terms of the components of the system of Fig. 1 as follows: In the set of Eq. (2), is the high cut-off time, the low cut-off time, the time-delay, x the dimensionless dynamical variable of the system, n o = 4 the number of junction diodes in series and whose characteristics are the thermal voltage V o = 25 mV and inverse saturation current i o = 5 μ A. u is the cubic-nonlinear coefficient. V rf , V dc , and V B are the radio-frequency (rf), the direct-current (dc) half-wave voltages, and the bias voltage of the Mach-Zehnder modulator, respectively. The gain of the (1) amplifier is G, the sensitivity of the photodiode S, and P is the power of the laser diode. The values of these parameters are taken compatible with the experiment (Talla Mbé et al. 2019; Kamaha et al. 2020): V rf = 3.8 V, V dc = 5 V, R = 2.5 kΩ, r = 300 Ω , L = 0.1 mH, C 1 = 270 pF, C 2 = 9.15 nF, S = 4.75 V/mW. When using OEOs for sensings, detections, and measurements, the oscillation limitcycle frequency plays a fundamental role since the principles are based on the variation of this frequency recorded on an oscilloscope . It was demonstrated that due to the cubic-nonlinear term u ∫ where f sd is the oscillating frequency of the standard OEO, = u 3 the cubic-nonlinear parameter, and = the cut-off times ratio. For a delay line of 656 m, f sd = 304.8 kHz. Considering the following variable y = ∫ t 0 x ds, y st denotes the corresponding fixed point of our system. It is important to note that the CNOEO is identified by ≠ 0 , whilst for the standard OEO, = 0 . Experimentally, can be turned through the gain G of the amplifier [see Eq. (2)]. The next sections (Sects. 3, 4 and 5) deal with some sensings and one measurement using the CNOEO.

Circuit and principle of temperature sensing using a CNOEO
The setup for temperature sensing is given in Fig. 2. As in Zou et al. (2016), Zhu et al. (2014), the single-mode fiber (SMF) is the temperature sensor. Indeed, the SMF is In OEOs, the length l and refractive index n of the SMF are the two parameters that vary with the temperature. Differentiating Eq. (5) with respect to n and l, and assuming that l op >> l 1 , the variation of the frequency Δf due to heating/cooling yields: In Eq. (7), Δf is the variation of the frequency recorded by the oscilloscope, ΔT the variation of the temperature, 1 the expansion coefficient, and the thermo-optic coefficient of the SMF (Zhu et al. 2014). ΔT also characterizes the temperature sensitivity and precision; small values of ΔT means that our system can detect a slight temperature change. Figure 3 shows the variation of the temperature ΔT against the variation of the recorded frequency in the oscilloscope Δf [see Eq. (7)]. Figure 3 presents 5 plots obtained by varying . Each plot of Fig. 3 is a linear variation with a positive slope which decreases as the cubic-nonlinear parameter increases. From up to down, the curves are obtained for = 0, = 2.0 × 10 −9 , = 1.48 × 10 −8 , = 4 × 10 −7 , and = 1.39 × 10 −4 . It can be noticed that for a given variation of the frequency recorded in the oscilloscope, the value of the change in temperature sensed by the SMF decreases with the increase of the cubic-nonlinear parameter (see Fig. 3). Some cases are illustrated in Table 1. For instance, for a recorded value of Δf = 1050 Hz, the standard OEO displays a variation of the temperature of ΔT = 229.6 • C which is less precise compared to ΔT = 159.3 • C, ΔT = 110.5 • C, ΔT = 60.58 • C, and ΔT = 22.24 • C obtained with the CNOEO when the cubic-nonlinear parameter is increasingly monitored to the values = 2.0 × 10 −9 , = 1.48 × 10 −8 , = 4 × 10 −7 , and = 1.39 × 10 −4 , respectively. Noting that the higher precision (smaller value of ΔT ) is obtained with the higher value of ; that is (4) = op + 1 ,  Table 1). Figure 4 displays the variation of the temperature against the length (l). It is also the plots of 5 graphs obtained for different values of as that shown in Fig. 3. But, it is an inverse variation (as Eq. (7) demonstrates) which values decrease as the cubic-nonlinear parameter is being increased (see Fig. 4). Table 2 shows one numerical illustration.

Results of temperature variation sensing
Here, one still observes the same sensitivity of ΔT as increases. For example, when l is equal to 1000 m, ΔT = 288.6 • C for the standard OEO ( = 0 ), whereas it is significantly

Circuit and principle of refractive index sensing using a CNOEO
Based on Wang et al. (2014), the proposed setup used for the refractive index sensing is shown in Fig. 5. The liquid whose refractive index n x is to determine is inserted in the glass cell of diameter d x . The light from the optical delay line passes through the first free space, then the glass cell, and after, the second free space. It hits the target and is reflected again to the same path. The extra time-delay due to the liquid will cause a variation of the oscillation frequency recorded on the oscilloscope. This oscillation frequency will be used to evaluate the refractive index n x . Indeed, the total time-delay of the signal to come to completion is the sum of the electrical time-delay ( e ) and the optical time-delay ( opt ) given by Pham et al. (2014):  Fig. 5 Setup of the refractive index sensing using a CNOEO. l a1 is the first free space distance of refractive index n a1 , l a2 the second free space distance of refractive index n a2 , d x represents the diameter of the glass cell of refractive index n x Let us note the length of the free space l a = l a1 + l a2 with the refractive index n a = n a1 = n a2 , which is the refractive index of air. The mathematical expression of opt is given below: where l 0 is the total length of the optical delay line in Fig. 5, n 1 its refractive index, and c the speed of light in the vacuum. In Eq. (9), 2 accounts for the round trip of light in the media of refractive indexes n a1 , n x , and n a2 (see Fig. 5). Equation (8) becomes (Nguyen et al. 2010): Let us consider n 1 L 1 = n 1 l 0 + c e . It implies that Eq. (10) becomes By definition, the oscillation frequency f osc = k , with k an integer standing for the mode number ( k = 1 for the fundamental mode) (Pham et al. 2014;Nguyen et al. 2010). Thus, Differentiating Eq. (12), the variation of frequency is given by: Equation (13) is similar to the one obtained by Nguyen et al. (2010). Here, FSR is the free spectral range which expression is given by: After the substitutions of FSR and Eq. (3) in Eq. (13), one obtains the mathematical expression of the variation of the refractive index given below: Finally, the refractive index n x of the liquid is computed through: 4.2 Results of refractive index variation sensing. Figure 6 plots the variation of the refractive index against the optical length ( l 0 ). It can be seen that the variation of the refractive index Δn x reduces as one keeps on increasing for a given value of optical length l 0 . This variation is very small since it is in the order Δf osc n x = n a + Δn x of five digits showing then the high sensitivity of our system. That sensitivity is more accurate when the cubic-nonlinear parameter becomes more important. From up to down, the curves are obtained for = 0 (standard OEO), = 2.0 × 10 −9 , = 1.48 × 10 −8 , and = 1.39 × 10 −4 . In Fig. 7, we also show the variation of the refractive index Δn x against the variation of the frequency Δf osc [see Eq. (15)]. Here also, one remarks that with higher values of the cubic-nonlinear parameter , higher precision Δn x in the values of the refractive index are obtained. Moreover, Δn x is reduced to the order of 8 digits. 5 Distance measurement using a CNOEO

Circuit and principle of distance measurement using a CNOEO
When using an OEO for distance measurement, the distance to evaluate l a is the length within the OEO and a target, as shown in Fig. 8 (Zou et al. 2016;Wang et al. 2014). The principle is as follows: the light from the optical delay line travels through the free space to the target. The target reflects the light to regain the optical path of the CNOEO. This induces an additional time-delay with n a and l a being the refractive index of air and the distance to measure, respectively. As a result, the oscillation frequency changes and is read by the oscilloscope. The total timedelay of the system of Fig. 8 is the sum of the reference time-delay ( R ) and the additional time-delay ( A ) given by Wang et al. (2014): where with f R representing the reference frequency. Substituting Eq. (16) and (18) in Eq. (17) gives Wang et al. (2014): From Eq. (19) and applying = 1 f osc , l a is given as follows:  Fig. 8 Setup of the distance measurement using CNOEO. l a is the distance to determine 311 Page 12 of 13

Conclusion
In this work, we have carried out a comparative study of temperature sensing, refractive index sensing, and distance measurement using both the standard and the cubic-nonlinear optoelectronic oscillators. For each of these sensings and the measurement, we have proposed an appropriate setup as well as the analytical expression of the sensing/measurement quantity. It globally comes out that the cubic-nonlinear optoelectronic oscillator (CNOEO) is more sensitive; that is higher precisions in the sensings and the measurement are achieved with the CNOEO than using the standard optoelectronic oscillator. Such precisions become sharper as the values of the cubic-nonlinear parameter become larger. With the CNOEO, results that span several orders of magnitude have been recorded compared to those with the standard OEO. Further investigations will focus on using other novel architectures of OEO (Chengui et al. (2016), Chengui et al. 2018, Kouayep et al. (2020, Nguewou-Hyousse and Chembo (2020)) for other metrological quantities such as strain, load, angular velocity, and more.