Analytical Analysis and Experimental Validation of a Multi-parameter Mach-Zehnder Fiber Optic Interferometric Sensor

Here we report a simple analytical technique to model a Mach–Zehnder fiber optic interferometric sensors that allow us to predict and calculate via computer simulations parameters that are not easily obtained experimentally. This model was calibrated and compared with experimental data using a 120 mm sensor for measurements of temperature, refractive index and water level. For instance, we were able to calculate the effects on the cladding effective index caused by the variation of those physical parameters. Moreover, this analysis could further our understanding of such sensors and allow us to make predictions about their use in different applications and even their behavior with different sensing lengths.


I. INTRODUCTION
The use of fiber optic sensors is in considerable growth.Their applications range from structural health to ecological monitoring.Since the last decades, these sensors have been widely used in industries due to their enormous advantages such as high sensitivity, precision, immunity to electromagnetic fields, and safe implementation and operation.Within range of optical sensors, interferometric techniques stand out for usually allowing greater sensitivity than traditional ones.
Several interferometric sensors have been published based on the traditional principles of Mach-Zehnder, [1], [3], [5]- [7].It has been widely seen that for each sensor constructed, values such as their sensitivity and operational range is often highly dependent on the sensors length.This means that to find an analytical or numerical model that can accurately describe the functioning of a specific type of sensor is imperative in order to appropriately project new sensors with certain characteristics.The wavenumbers  and  are values that depend on the beams wavelengths  e  as defined by ( 3) and ( 4), which are functions of the refractive index of the core and cladding, respectively.The method used to relate  with the refractive index , is according to the already known ( 5) and ( 6), where ,  and  represent the wave propagation speed, frequency, and speed of light in vacuum, respectively.

𝛽 =
(3) 531 The interrogation performed in [1] is based on the shift in wavelength of peaks and valleys generated in the spectrum after the combination of both signals traveling in the cladding and core.
This happens because when the external medium characteristics are changed, the effective refractive index of the cladding is changed, altering the value of  but maintaining constant the value of  .
Thus, from analyzing the experimental data obtained in [1], we can calculate precisely what is the corresponding change of the cladding effective refractive index as a function of the external medium liquid level, temperature, and its refractive index.In order to simulate the broadband optical source used in [1] as the sensor input to generate the spectrum obtained after the sensor when combining both beams, we use as input different wavelengths for the electric fields  and  ranging from 1500 nm to 1580 nm.Using the parameters defined in [1] of z = 12 cm, core refractive index ( ) = 1.4565 and cladding refractive index ( ) = 1.45 we can obtain the optical spectrum shown in Fig. 2. As already mentioned, the studied sensor uses the variation of the cladding effective refractive index, which results in the displacement of peaks and valleys of the interferometric spectrum, in order to perform its measurements.Thus, in our simulations we vary the effective refractive index of the cladding ( ) and we compare the obtained wavelength shift of a given peak with the data obtained from [1] as a function of the external medium level, temperature and refractive index.

A. Liquid's Level Sensing Simulation
According to the performed calibration, the initial cladding refractive index that correctly feats the peaks obtained from the simulation with the experimental data is  = 1.4500023.In order to reach the wavelength variation of the peak initially centered in 1541.05nm, as it was chosen in [1], that occurs when filling of the container with 120 mm of water, the cladding effective refractive index was 532 varied until the final value of  = 1.4499992.The simulation was done for steps of the cladding effective refractive index of 10 refractive index units (RIU), resulting in 31 points and a negative total change of ∆ = −3.110RIU.Each interaction leads to a spectral shift to the left, as it was also seen in [1].Fig. 3 shows the simulation spectra obtained for the peak initially centered at 1541.05 nm as function of the water level.As it can be seen, when the sensor is completely filled with water (120 mm) the central wavelength shifts to 1540.32 nm which corresponds to a variation of the cladding effective index of −3.110 RIU.From the successive positions of the displaced peak shown in Fig. 3, we were able to compare our simulation results with the experimental data measurements, which is shown in Fig. 4, where the central wavelength of the peak initially centered is plotted against the water level variation.This result highlights the fact that our simple approach of modeling our sensor only as a sum of two different optical paths can give a very accurate prediction of the sensor behavior, avoiding the 533 necessity of performing a more complete modal analysis which is much more time consuming and computationally complex.
Finally, in Fig. 5 it can be seen how the variation of water level affects the actual effective refractive index of the cladding.It can be observed the variation is linear with a rate of -2.58 x 10 -8 RIU/mm.It is important to note that an increase in liquid level actually decreases the effective refractive index of the cladding.Moreover, it is clear that even such small variation in effective refractive index of -2.58 x 10 -8 RIU/mm can give rise to a moderate sensor sensitive when analyzing in terms of wavelength shift, which is 6 pm/mm.

B. Liquid Temperature Sensing Simulation
Analogous to the liquid level measuring, for temperature sensing simulation we used the same method already described.Nevertheless, some points in this simulation should be highlighted.To measure the temperature, the experimental data [1] varied the temperature of the water with the pipette where the sensor was placed with the 120 mm completely full.This means that the initial cladding effective refractive index is not the same as the one in the previous topic.For this procedure, we calibrated the refractive index to have the same value of  = 1.4565 but a value  = 1.4499987.
Note that this value is very close to the one found for the filled vessel in the liquid level sensing simulation.The difference of 510 RIU is attributed to the fact that the level experiment was done when the temperature was 25°C degrees and here the simulation and the temperature experimental data variations start at 22°C.
In order to achieve the wavelength variation that occurs according to the liquid temperature change from 22°C to 55°C, the cladding effective refractive index was varied to the final value of  = 1.4500043.As in the previous topic, the value of each step for the variation of the effective refractive index was 10 RIU, resulting in 58 interactions and a positive variation of ∆ = 5.610 RIU.
Due to the positive variation, this time the spectral shift occurred to the right.

534
The central wavelength of the peak initially centered at 1540.2 nm as a function of temperature can be seen in Fig. 6.Again we see that our simplified approach of modeling the sensor as only the sum of two electric fields traveling through two different optical paths describes the sensor behavior very accurately.The variation of the cladding refractive index as a function of temperature is shown in Fig. 7, where again we obtain a linear relation, but now presenting a positive inclination and a sensitivity of 1.69710 RIU/°C.This positive inclination in Fig. 7 means that increasing the water temperature caused an increase in effective refractive index of the cladding, which leads to a red shift in the central peak wavelength as it was obtained also experimentally.Again, it can be observed that even a very small sensitivity of 1.69710 RIU/°C can lead in terms of spectra wavelength variation to a moderately high sensor sensitivity of 39 pm/°C.Journal of Microwaves, Optoelectronics and Electromagnetic Applications, Vol.17, No. 4, December 2018 DOI: http://dx.doi.org/10.1590/2179-10742018v17i41540537 than 5 cm, which would allow a very big visualization window around each peak to analyze its variation as a function of external physical parameters.However, this result shows that the use of sensors that are longer than a few tens of centimeters as optical sensing devices can be problematic since a small FSR means that consecutive peaks are very close together, which would make necessary the use of a very small wavelength window around a given peak to analyze its variation.Finally, again for sensor lengths from 1 to 12 cm, the effective refractive index of the cladding was varied simulating exactly the changes in water level, temperature and refractive indexes used in the previous sections.The sensitivity of the peak closest to 1541 nm for variations of water level obtained as a function of the sensor length can be seen in Fig. 12.It was observed that the sensitivity of the sensor remains constant, with the value of 6 pm/nm regardless of the sensor length chosen.Similar results were obtained when analyzing the sensor sensitivity to temperature and refractive index, that also remained constant with the values of 39 pm/°C and 8.8 nm/RIU, respectively, for any given sensor length.Nevertheless, it is important to highlight the fact that, our analytical analysis is based on a simplified model considering only one average cladding mode.Therefore, sensors experimentally built with other lengths may exhibit sensitivity variations due to factors that are not addressed in this model, such as the fact that some cladding modes might propagate in the fiber for longer distances then others, being relevant only for sensors up to a certain length.IV.CONCLUSION In conclusion, we presented a simple analytical model that describes the behavior of an all-fiber optic interferometric sensor that allow us to simulate its usage for liquid level sensing, liquid temperature sensing, and liquid refractive index sensing.In this paper, we are also able to formulate a deeper understanding of how those external physical parameters affect the cladding effective refractive index, showing for each case its sensitivity and variation rate.This simple analytical formulation was compared and validated with experimental data showing that more complete modal analysis is not needed in order to understand and predict the functioning of such sensors.This understanding can be applied to project similar sensors of other lengths and also to simulate the use of a similar structure for other applications.Those results could potentially be extended to many other Mach-Zehnder type fiber optical sensor.

2 )Fig. 1 .
Fig. 1.Fiber arrangement using an off-setted fiber with reduced core to generate the cladding modes.a) 3-D view, b) longitudinal section, c) sensor schematic used in the experiments.

Fig. 3 .
Fig. 3. Spectrum corresponding of the peak initially centered at 1541.05 nm as a function of water level.

Fig. 4 .
Fig. 4. Central wavelength of the peak initially at 1541.05 nm as a function of the liquid level.

Fig. 5 .
Fig. 5. Variation of cladding effective refractive index as a function of the liquid level.

Fig. 6 .Fig. 7 .
Fig. 6.Central wavelength of the same peak as a function of the temperature.

Fig. 11 .
Fig. 11.Free spectral range as a function of the sensor length.

Fig. 12 .
Fig. 12. Sensor sensitivity for water level variations as a function of the sensor length.